We give a treatment of relative Calabi–Yau structures on functors between R-linear stable ∞-categories, with R any 𝔼∞-ring spectrum, generalizing previous treatments in the setting of dg categories. Using their gluing properties, we further construct relative Calabi–Yau structures on the global sections of perverse schobers, that is, categorified perverse sheaves, on surfaces with boundary. We treat examples coming from Fukaya categories and representation theory. In a related direction, we define the monodromy of a perverse schober parametrized by a ribbon graph on a framed surface and show that it forms a local system of stable ∞-categories.
Mathematics Subject Classification 202018N60KeywordsCalabi–Yau categoriesrelative Calabi–Yau structureperverse schoberstopological Fukaya categoriesEuropean Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie101034255The author acknowledges support from the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC 2121 ‘Quantum Universe’ – 390833306. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 101034255. The author is a member of the Hausdorff Center for Mathematics at the University of Bonn (DFG GZ 2047/1, project ID 390685813).Introduction
Let k be a field. A k-linear triangulated category C with finite-dimensional Homs is called n-Calabi–Yau if there exists an isomorphism of vector spaces
ExtCi(X,Y)≃ExtCn−i(Y,X)∗,
bifunctorial in X,Y∈C. To obtain a well-behaved notion of n-Calabi–Yau structure on a proper k-linear stable ∞-category 𝒞, one can ask for a trivialization S≃[n] of the Serre functor S, that is, the functor satisfying the following duality for k-linear derived Homs in 𝒞:
Mor𝒞(X,Y)≃Mor𝒞(Y,S(X))∗∈𝒟(k).
The trivialization S≃[n] is called a weak right n-Calabi–Yau structure on 𝒞. Note that the natural transformations Mor(id𝒞,S) describe the dual Hochschild homology HH(𝒞)∗ of 𝒞. The identification S≃[n] may thus additionally be required to be S1-invariant, that is, to lie in the image of a dual cyclic homology class under the morphism HHS1(𝒞)∗→HH(𝒞)∗. This leads to the notion of a right n-Calabi–Yau structure on 𝒞. The importance of this S1-invariance comes from the relation with topological field theories [12, 32].
There is a similar notion of left n-Calabi–Yau structure on a smooth k-linear stable ∞-category 𝒞, where instead of the Serre functor, one asks for a trivialization of the bimodule left dual id𝒞! of the evaluation bimodule. The endofunctor id𝒞! is sometimes called the inverse dualizing bimodule, as it is the inverse of the Serre functor if 𝒞 is smooth and proper.
In this paper, we will be concerned with generalizations of left and right Calabi–Yau structures to R-linear functors between R-linear stable ∞-categories, where R is a base 𝔼∞-ring spectrum. Calabi–Yau structures on functors are also referred to as relative Calabi–Yau structures. This notion was suggested by Toën [43] and fully worked out in the setting of dg categories by Brav–Dyckerhoff [2]. The first half of this paper concerns a careful lift of this theory to the setting of R-linear stable ∞-categories.
There are many natural examples of relative Calabi–Yau categories, the known ones usually come from Fukaya categories, representation theory, topology and algebraic geometry. Applications of relative Calabi–Yau structures include the constructions of shifted symplectic structures and Lagrangian structures on moduli spaces of objects [3], the construction of additive categorifications of cluster algebras with coefficients [8, 18, 45], of 2-Calabi–Yau exact ∞-categories/extriangulated categories from right 2-Calabi–Yau functors [8] and of framed E2-algebra structures on Hochschild cohomology [4], to name a few.
Relative Calabi–Yau structures possess the remarkable feature that they can be glued together along suitable pushouts or pullbacks of categories to produce new relative or absolute Calabi–Yau structures. As emphasized by Brav–Dyckerhoff [2], Calabi–Yau structures should be considered as noncommutative orientations and their gluing property as a noncommutative version of the gluing property of oriented manifolds with boundary along boundary components.
The gluing properties of relative Calabi–Yau structures can be used to construct relative Calabi–Yau structures on functors with target the topological Fukaya categories associated with framed marked surfaces [2]. These topological Fukaya categories can be seen as the global sections of perverse schobers on surfaces, that is, categorified perverse sheaves in the sense of [26]. More generally, perverse schobers allow us to define Fukaya categories of surfaces ‘with coefficients’. Using the framework of [5], we parametrize such perverse schobers by a ribbon graph homotopic to the surface. More specifically, such a parametrized perverse schober then amounts by definition to a certain constructible sheaf of stable ∞-categories on the ribbon graph, which is locally described by spherical adjunctions.
In the second half of this paper, we discuss ways to construct relative Calabi–Yau structures on the global sections of more general perverse schobers on surfaces. The main results of this second half can be summarized as follows:
We associate a local system of stable ∞-categories to a parametrized perverse schober on a framed marked surface encoding its monodromy on the surface away from its singularities. We also establish the independence of this local system on the chosen parametrizing ribbon graph. Further, we prove that parametrized perverse schobers without singularities are determined, up to equivalence, by their monodromy local systems.
In the special case of perverse schobers without singularities, we prove the existence of a relative Calabi–Yau structure on its ∞-category of global sections, given the invariance of a local (negative) cyclic homology class under the monodromy action. This generalizes Brav–Dyckerhoff’s result [2] on relative Calabi–Yau structures on topological Fukaya categories of framed surfaces.
We describe conditions which guarantee the existence of relative Calabi–Yau structures on the local and on the global sections of singular parametrized perverse schobers.
We apply our results to construct relative Calabi–Yau structures on classes of examples, including Fukaya–Seidel categories, periodic topological Fukaya categories of marked surfaces, the derived categories of relative Ginzburg algebras associated with n-angulated surfaces, as well as variants of the latter which are linear over an arbitrary 𝔼∞-ring spectrum.
The remainder of the introduction is structured as follows. We begin in Section 1.1 by reviewing the notion of a relative Calabi–Yau structure over a base 𝔼∞-ring spectrum. We proceed in Section 1.2 with describing our results on relative Calabi–Yau structures on perverse schobers. In Section 1.3, we describe the implications of our results for Fukaya–Seidel categories and other Fukaya-type categories.
Relative Calabi–Yau structures
Let R be the base 𝔼∞-ring spectrum. Our setting for the definition of relative Calabi–Yau structures is that of stable, presentable, dualizable, R-linear ∞-categories and dualizable (in particular colimit-preserving), R-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-linear Hochschild homology, as well as its S1-action, which we obtain from the formalism of traces [23, 44].
Consider a dualizable R-linear functor F:𝒟→𝒞 between dualizable R-linear ∞-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-linear ∞-categories. The R-linear ∞-category 𝒞 being proper means that the evaluation functor ev𝒞:𝒞∨⊗𝒞→RModR admits an R-linear right adjoint, which can be identified with an endofunctor id𝒞∗ of 𝒞. If 𝒞 is compactly generated, the functor id𝒞∗ is a Serre functor on 𝒞. The natural transformations between id𝒞∗ and the identity functor are described by the dual Hochschild homology HH(𝒞)∗. In a similar way, a class σ:R[n]→HH(𝒟,𝒞)∗≔cof(HH(𝒞)∗→HH(𝒟)∗) in the dual relative Hochschild homology of F defines a map α:id𝒞→id𝒞∗[1−n] together with a null-homotopy of the composite map id𝒟→id𝒟∗[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 σ a weak right n-Calabi–Yau structure on F if the vertical maps in the above diagram are equivalences. A right n-Calabi–Yau structure on F then further consists of a lift of σ to a relative dual cyclic homology class. If F admits a right n-Calabi–Yau structure, we also say that 𝒟 is relative right n-Calabi–Yau. Non-relative right Calabi–Yau structures correspond to the case 𝒞=0.
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→B admits a weak Calabi–Yau structure if and only if the k-linear functor 𝒟(f):𝒟(A)→𝒟(B) between the compactly generated derived ∞-categories admits a weak Calabi–Yau structure. The possibility to consider relative Calabi–Yau structures on dualizable k-linear ∞-categories thus makes the ∞-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-Calabi–Yau, we have a trivialization id𝒟∗[−n]≃id𝒟 of the shifted Serre functor. If instead 𝒟 is relative right Calabi–Yau, we have some natural transformation id𝒟∗[−n]→id𝒟, together with an identification of its cofiber with F∗(id𝒞)=GF, where G is the right adjoint of F. To get a well-behaved theory, it is however important that this is not just any identification of the cofiber of id𝒟∗[−n]→id𝒟 with GF, but rather that there is a specific such cofiber sequence coming from a relative dual Hochschild class.
Left Calabi–Yau structures for functors between smooth R-linear ∞-categories are defined similarly by replacing the right adjoint of the evaluation functor by the left adjoint, corresponding to an endofunctor id𝒞!, 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 ∞-categories. There are also variants of this for (∞,2)-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-linear ∞-categories and dualizable functors.
Theorem 1
(Theorem 3.15, [2] for R=k a field). If the functors ℬ1×ℬ2→𝒞1 and ℬ2×ℬ3→𝒞2 carry R-linear left n-Calabi–Yau structures, which are compatible at ℬ2, then the functor ℬ1×ℬ3→𝒟 inherits an R-linear left n-Calabi–Yau structure.
For the gluing of right Calabi–Yau structures, we consider a pullback diagram of proper dualizable R-linear ∞-categories and dualizable functors as follows.
Theorem 2
(Theorem 3.16). If the functors 𝒞1→ℬ1×ℬ2 and 𝒞2→ℬ2×ℬ3 carry R-linear right n-Calabi–Yau structures, which are compatible at ℬ2, then the functor 𝒟→ℬ1×ℬ3 inherits an R-linear right n-Calabi–Yau structure.
We note that the analogue of Theorem 2 was not previously known for right Calabi–Yau structures on dg categories [2].
Perverse schobers and relative Calabi–Yau structures
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 ∞-categories defined on a spanning ribbon graph embedded in the surfaces. Concretely, a perverse schober ℱ parametrized by a graph G is encoded as a functor ℱ:Exit(G)→St. Here St denotes the ∞-category of stable ∞-categories, and the domain denotes the exit path category of G, whose objects are the vertices and edges of G and whose morphisms describe the incidence between vertices and edges. The limit of this functor, denoted by Γ(G,ℱ), is called the ∞-category of global sections of ℱ.
We remark that the usage of enhanced triangulated categories (such as the stable ∞-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 ∞-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 G-parametrized perverse schober ℱ: the value of ℱ at any edge of G is independent of the chosen edge, up to equivalence, and should be considered as the generic stalk. Denote the generic stalk by 𝒩. Locally at each vertex v of Γ, a perverse schober is described by a spherical adjunction 𝒱↔𝒩, with 𝒱 called the ∞-category of vanishing cycles at v. If 𝒱≠0, we call the vertex v a singularity of ℱ.
Let G0 be the set of vertices of G. Given a perverse schober ℱ with a set of singularities P⊂G0, we wish to associate a local system valued in St on S\P, which we describe as a group homomorphism
π1(S\P)⟶π0Aut(𝒩)
to the group of equivalence classes of autoequivalences of the generic stalk 𝒩.
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 Fr(S\G0)→S\G0. The fiber of the frame bundle has the homotopy type of the circle S1, the monodromy of the local system along the fiber is given [2]. Suppose now that we choose a framing ξ of the surface S\P, meaning a section of its frame bundle. We can then pull back to a local system ξ∗ℒℱ on S\G0, and crucially, this local system extends to S\P. This defines the desired monodromy local system of ℱ. We stress that this local system depends on the choice of framing ξ.
Note that in the special case that 𝒩 is 2-periodic, that is, [2]≃id𝒩, the local system on the frame bundle has trivial monodromy on the fiber. It thus already reduces to a local system on S\P, and no choice of framing is required as input.
Perverse schobers without singularities are fully determined by their monodromy.
Proposition 1
(Proposition 4.34). Let ξ be a framing of S. Let ℱ1,ℱ2 be two G-parametrized perverse schobers without singularities with identical generic stalk 𝒩. Then there exists an equivalence ℱ1≃ℱ2 if and only if the corresponding local systems
ξ∗ℒℱ1,ξ∗ℒℱ2:π1(S\P)⟶π0Aut(𝒩)
are equivalent.
The notion of a non-singular parametrized perverse schober is thus non-canonically equivalent to the notion of a local system of stable ∞-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 ∞-category 𝒩: without further assumptions on 𝒩, 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 ∞-category of global sections describes the desired 𝒩-valued topological Fukaya category. In the case 𝒩=𝒟(k), this ∞-categorical topological Fukaya category recovers the derived ∞-category of the dg categorical topological Fukaya category or equivalently of the A∞-categorical partially wrapped Fukaya category. If 𝒩 is 2-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-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-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 ∞-category of right modules over the 2-periodic sphere spectrum by Lurie [33].
Local Calabi–Yau structures. Let ℱ be an R-linear G-parametrized perverse schober. Locally at any vertex v of the graph G, with incident edges e1,…,em, the perverse schober ℱ is given by a collection of functors ℱ(v)→ℱ(ei)≃𝒩, 1≤i≤m. One can show that these functors arise, up to suitable equivalence, from a single spherical adjunction F:𝒱↔𝒩:G via an explicit construction based on the relative Waldhausen S•-construction (see Proposition 4.10). We say that the adjunction F⊣G underlies ℱ at v.
Proposition 2
(Combine Propositions 4.10 and 5.2). Let ℱ be a G-parametrized perverse schober, with underlying spherical adjunction F:𝒱↔𝒩:G near a vertex v of G.
IfFadmits a rightn-Calabi–Yau structure, which restricts to a right(n−1)-Calabi–Yau structure on𝒩, then the functorℱ(v)→∏i=1nℱ(ei)also admits a rightn-Calabi–Yau structure.
IfGadmits a leftn-Calabi–Yau structure, which restricts to a left(n−1)-Calabi–Yau structure of𝒩, then the right adjoint∏i=1nℱ(ei)→ℱ(v)of the above functor also admits a leftn-Calabi–Yau structure.
We further prove a novel criterion for a spherical functor F:𝒱→𝒩 between compactly generated, proper R-linear ∞-categories, where 𝒩 is weakly right (n−1)-Calabi–Yau, to admit a weak right n-Calabi–Yau structure: this is the case if and only if its twist functor T𝒱 is equivalent to the shifted Serre functor id𝒱∗[1−n] (see Proposition 5.9).
Global Calabi–Yau structures. Given a G-parametrized perverse schober ℱ, we can evaluate global sections at the external (i.e., boundary) edges of G, whose set is denoted by G1∂. This yields a functor
∏e∈G1∂eve:Γ(G,ℱ)⟶∏e∈G1∂ℱ(e).
The right adjoint of this functor is denoted by ∂ℱ.
Typically, a relative Calabi–Yau structure on the ∞-category of global sections Γ(G,ℱ) arises in the smooth setting as a left Calabi–Yau structure on the functor ∂ℱ and in the proper setting as right Calabi–Yau structure on the functor ∏e∈G1∂eve.
Note that finite limits of proper dualizable R-linear ∞-categories in St (or equivalently in the ∞-category LinCatR of R-linear ∞-categories) are not necessarily again dualizable and proper. We can fix this issue by forming the limits in the ∞-category LinCatRdual of dualizable R-linear ∞-categories (see Corollary 3.14). The arising notion of ∞-category of global sections is denoted by Γdual(G,ℱ); we call these the dualizable global sections. In the proper setting, we should thus ask for the restriction of ∏e∈G1∂eve to Γdual(G,ℱ) to be relative right Calabi–Yau.
When the global sections describe the partially wrapped Fukaya category of a surface, the difference between Γ(G,ℱ) and Γdual(G,ℱ) can be explained as follows: the category Γ(G,ℱ) describes the usual smooth partially wrapped Fukaya category of the marked surface. The proper full subcategory Γdual(G,ℱ)⊂Γ(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
(Theorem 5.8). Let ℱ be a G-parametrized perverse schober without singularities valued in dualizable R-linear ∞-categories. Suppose that the generic stalk 𝒩 is smooth and admits a left (n−1)-Calabi–Yau structure
η:R[n−1]→HH(𝒩)S1.
Suppose that the monodromy local system HH(ℒℱ)S1 on S preserves η. Then the functor
∂ℱ:∏e∈G1∂ℱ(e)⟶Γ(G,ℱ)
admits a left n-Calabi–Yau structure.
A similar statement holds for relative right Calabi–Yau structures on the dualizable global sections Γdual(G,ℱ).
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.
Examples: Fukaya categories and Fukaya-type categories
Fukaya–Seidel categories. Let X be an exact symplectic manifold of dimension 2n and π:X→𝔻 a Lefschetz fibration with base the disc. Let F be the regular fiber of π and Fuk(F) the proper Fukaya A∞-category of compact Lagrangians in F. The Fukaya–Seidel A∞-category FS(π) is equivalent to the directed A∞-subcategory of 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-linear stable ∞-category 𝒟(FS(π)).
The formalism of parametrized perverse schobers on the disc 𝔻, considered as a marked surface with a single marked point, realizes the derived Fukaya–Seidel category 𝒟(FS(π)) as the global sections of a perverse schober. The generic stalk of the schober is the derived Fukaya category of the fiber 𝒟(Fuk(F)). 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 Fuk(F) given by the vanishing cycles. The ribbon graph parametrizing ℱ is chosen so that the singular values all lie at 1-valent vertices. There is a further non-singular (m+1)-valent vertex v, with m the number of vanishing cycles. The value of ℱ at v is the directed ∞-category ℱ(v)≃Fun(Δm−1,𝒟(Fuk(F))).
Any spherical object in a weak (n−1)-Calabi–Yau category gives rise to a weak n-Calabi–Yau spherical functor (see Lemma 6.4). The gluing properties of right Calabi–Yau structures thus yield a weak relative right n-Calabi–Yau structure on 𝒟(FS(π)). This induces the known natural transformation [38] from the Serre functor id𝒟(FS(π))∗→id𝒟(FS(π))[n+1].
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-Calabi–Yau structure but also a weak relative left n-Calabi–Yau structure.
Theorem 4
(Theorem 6.1). Let π:X→ℂ≥0 be a Lefschetz fibration as in Section 6.1.
The derived∞-category of the Fukaya–Seidel category𝒟(FS(π))arises as the∞-category of global sections of the perverse schoberℱon the disc from Construction6.3.
The smooth and proper derived Fukaya–Seidel category𝒟(FS(π))admits both a weak relative leftn-Calabi–Yau structure and a weak relative rightn-Calabi–Yau structure.
The Fukaya–Seidel category is defined in [20] as a partially wrapped Fukaya category with a stop in the fiber over ∞. 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-Calabi–Yau structures on 1-periodic topological Fukaya categories of marked surfaces. These can be considered as ℤ/1ℤ-graded versions of the partially wrapped Fukaya categories. Their construction is the topic of Section 6.2. Relative right 2-Calabi–Yau structure induces 2-Calabi–Yau Frobenius exact ∞-structures (see [8]), which in turn induce 2-Calabi–Yau Frobenius extriangulated structures on the homotopy 1-categories. In the case of 1-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-linear ∞-categories, with k a field, these periodic categories are smooth but not proper, since the Ext-groups are non-zero in infinitely many degrees. This changes when we work with respect to a different base. For n an integer, the derived category of n-periodic chain complexes is equivalent to the derived ∞-category of the dg algebra k[tn±] of graded Laurent polynomials, with generator tn in degree n. If n is even, then k[tn±] is graded commutative and thus gives rise to an 𝔼∞-ring spectrum. If n is odd, we can consider k[tn±] as a k[t2n±]-linear algebra object. Over the base k[tn±], or k[t2n±] if n is odd, the derived ∞-category 𝒟(k[tn±]) is both smooth and proper and admits left and right n-Calabi–Yau structures.
Considering the n-periodic topological Fukaya category over the base k[tn±], or k[t2n±] if n is odd, Theorem 3 yields the desired relative (n+1)-Calabi–Yau structure on it.
Relative Ginzburg algebras over any base ring spectrum. The derived ∞-categories of relative Ginzburg algebras of n-angulated surfaces arise as the global sections of parametrized perverse schobers (see [5, 6]). In Section 6.3, we construct relative left n-Calabi–Yau structures on these derived ∞-categories. In the case n=3, this result is a special case of results [45, 46] on relative left 3-Calabi–Yau structures on relative Ginzburg algebras of ice quivers with potentials or equivalently relative Calabi–Yau completions. In the case n=3, these ∞-categories can further be expected to describe the derived ∞-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∗:𝒟(k)↔Fun(Sn−1,𝒟(k)):f∗, where f:Sn−1→∗ is the map from the singular simplicial set of the (n−1)-sphere to the point. The functor f∗ is left n-Calabi–Yau. Furthermore, the functor f¯∗, obtained by restricting f∗ to a functor 𝒟(k)→IndFun(Sn−1,𝒟perf(k)), is right n-Calabi–Yau.
We can replace 𝒟(k) by RModR, with R an 𝔼∞-ring spectrum, to obtain an R-linear version of this adjunction. We expect that both the Calabi–Yau structures of f∗ and f¯∗ can be lifted to the R-linear setting, but we only prove that we have a weak right Calabi–Yau structure on f¯∗. 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 f¯∗ yields weak relative right n-Calabi–Yau structures on the locally compact global sections of R-linear perverse schobers that generalize (Ind-finite) derived categories of relative Ginzburg algebras.
Besides the Calabi–Yau structures for this class of examples, many other classes of examples of R-linear relative Calabi–Yau structures also remain to be worked out.
Notation
We generally follow the notation and conventions of [31, 34]. In particular, we use the homological grading convention. Given an ∞-category 𝒞 and two objects X,Y∈𝒞, we denote by Map𝒞(X,Y) the mapping space. We denote the Ind-completion of 𝒞 by Ind(𝒞)=Indω(𝒞) and the subcategory of (ω-)compact objects of 𝒞 by 𝒞c. Given a functor F:𝒞→𝒟, we denote its left and right adjoints, if existent, by ladj(F) and radj(F), respectively.
Linear <inline-formula><alternatives><mml:math display="inline"><mml:mi mathvariant="normal">∞</mml:mi></mml:math><tex-math><![CDATA[\(\infty\)]]></tex-math></alternatives></inline-formula>-categories and Hochschild homology
In this section, we review background material on R-linear ∞-categories, with R an 𝔼∞-ring spectrum, different notions of duals of bimodules, smooth and proper R-linear ∞-categories and R-linear traces and Hochschild homology. Much of this material appears in a similar form in [2, 3, 23, 34, 35], though partly with less general proofs. The reader will find that most things work in the R-linear setting very similarly to those in the setting of dg categories.
Linear <inline-formula><alternatives><mml:math display="inline"><mml:mi mathvariant="normal">∞</mml:mi></mml:math><tex-math><![CDATA[\(\infty\)]]></tex-math></alternatives></inline-formula>-categories
Let Cat∞ be the ∞-category of ∞-categories and 𝒮 the ∞-category of spaces. We denote by 𝒫rL⊂Cat∞ the subcategory of presentable ∞-categories and left adjoint functors and by 𝒫rR⊂Cat∞ the subcategory of presentable ∞-categories and right adjoint functors. The ∞-category 𝒫rL admits a symmetric monoidal structure such that a commutative algebra object in 𝒫rL amounts to a symmetric monoidal presentable ∞-category 𝒞, satisfying that its tensor product -⊗-:𝒞×𝒞→𝒞 preserves colimits in both entries (see [34, § 4.8]). An example of a commutative algebra object in 𝒫rL is the ∞-category RModR of right module spectra over an 𝔼∞-ring spectrum R. Note that if R=k is a commutative ring, then RModk is equivalent as a symmetric monoidal ∞-category to the (unbounded) derived ∞-category 𝒟(k) (see [34, Thm. 7.1.2.13]).
Definition 2.1.
Let R be an 𝔼∞-ring spectrum. The ∞-category
LinCatR≔ModRModR(𝒫rL)
of modules in 𝒫rL over RModR is called the ∞-category of R-linear ∞-categories.
As noted in [35, § D.1.5], R-linear ∞-categories in the above sense are automatically stable. Given 𝒞∈LinCatR, we denote the result of the action of an element C∈RModR on X∈𝒞 by C⊗X∈𝒞.
Definition 2.2
([34, Def. 4.2.1.28]). Let R be an 𝔼∞-ring spectrum. Let 𝒞 be an R-linear ∞-category, and let X,Y∈𝒞. A morphism object is an R-module Mor𝒞(X,Y)∈RModR equipped with a map α:Mor𝒞(X,Y)⊗X→Y in 𝒞 such that for every object C∈RModR, the following composite morphism is an equivalence of spaces:
We thus have πiMor𝒞(X,Y)≃π0Map𝒞(X[i],Y) for all i∈ℤ.
Remark 2.3.
Morphism objects always exist, and the formation of morphism objects forms a functor
Mor𝒞(-,-):𝒞op×𝒞⟶RModR
which preserves limits in both entries (see [34, Rem. 4.2.1.31]).
The ∞-category LinCatR 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 ⊗. The tensor product of 𝒞,𝒟∈LinCatR arises as the geometric realization of the two-sided bar construction Bar(𝒞,𝒟)∗:Δop→𝒫rL, which is given informally by the formula Bar(𝒞,𝒟)n=𝒞⊗𝒫rLRModR⊗𝒫rLn⊗𝒫rL𝒟, where ⊗𝒫rL denotes the symmetric monoidal product of 𝒫rL. The symmetric monoidal ∞-category LinCatR is closed. As observed in [23, § 4.1], using that 𝒫rL is closed, the internal Hom in LinCatR, denoted LinR(𝒞,𝒟), 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 𝒫rL. We record the following functoriality of the internal Hom of LinCatR.
Lemma 2.4.
There is a functor
LinR(-,-):LinCatRop×LinCatR⟶LinCatR
satisfying
LinR(ℬ,LinR(𝒞,𝒟))≃LinR(ℬ⊗𝒞,𝒟),
functorial in ℬ,𝒞∈LinCatRop and 𝒟∈LinCatR. We call LinR(𝒞,𝒟) the R-linear ∞-category of R-linear functors from 𝒞 to 𝒟.
Proof.
To construct the functor LinR(-,-), we follow [34, Rem. 4.2.1.31]. Consider the functor
MapLinCatR(-⊗-,-):LinCatRop×LinCatRop×LinCatR⟶𝒮.
Moving the second copy of LinCatRop to the other side, we obtain a functor
LinCatRop×LinCatR→Fun(LinCatRop,𝒮),
whose image lies in the full subcategory of representable presheaves, since LinCatR is a closed monoidal ∞-category. Composing with the inverse of the Yoneda embedding LinCatR→Fun(LinCatRop,𝒮) yields the functor LinR(-,-). By construction, we have
functorial in 𝒜,ℬ,𝒞∈LinCatRop and 𝒟∈LinCatR. Composing again with the inverse of the Yoneda embedding shows (2.1), concluding the proof.∎
Lemma 2.5.
Let 𝒞,D∈LinCatR.
The forgetful functorLinR(𝒞,𝒟)→Fun(𝒞,𝒟)is exact and reflects finite limits and colimits.
The subcategory ofLinR(𝒞,𝒟)of functors admitting anR-linear right adjoint (i.e., of dualizable functors) is closed under finite limits and colimits.
Proof.
The forgetful functor LinR(𝒞,𝒟)→Fun(𝒞,𝒟) factors through the internal Hom FunL(𝒞,𝒟) in 𝒫rL, and the functor LinR(𝒞,𝒟)→FunL(𝒞,𝒟) is exact and reflects finite limits and colimits. Since FunL(𝒞,𝒟) and Fun(𝒞,𝒟) are stable ∞-categories, it thus suffices for part (1) to show that the full inclusion FunL(𝒞,𝒟)⊂Fun(𝒞,𝒟) preserves finite colimits and loops. Colimits of colimit-preserving functors again preserve colimits since colimits commute with colimits.
For part (2), it suffices to note that finite limit or colimit diagrams in LinR(𝒞,𝒟) turn into finite limit or colimit diagrams upon passing to right adjoints. By part (1), the property of the right adjoints to preserve colimits is preserved under such finite limits and colimits.∎
We fix an 𝔼∞-ring spectrum R. Recall that an R-linear ∞-category 𝒞∈LinCatR is called dualizable if it admits a duality datum consisting of evaluation and coevaluation functors
ev𝒞:𝒞∨⊗𝒞⟶RModR
and
coev𝒞:RModR⟶𝒞⊗𝒞∨,
satisfying the triangle identities.
Recall further that an ∞-category 𝒞 is called compactly generated if 𝒞≃Ind(𝒞c) is equivalent to the Ind-completion of its subcategory of compact objects.
Definition 2.6.
We denote by LinCatRdual⊂LinCatR the subcategory consisting of dualizable R-linear ∞-categories and dualizable functors, meaning R-linear functors whose right adjoint preserves colimits and is thus again R-linear.
We denote by LinCatRcpt−gen⊂LinCatR the subcategory consisting of compactly generated ∞-categories and compact object-preserving functors.
An R-linear ∞-category 𝒞∈LinCatR 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 ∞-category in the ∞-category 𝒫𝓇StL⊂𝒫rL of stable, presentable ∞-categories (see [35, Prop. D.7.3.1]).
In particular, any compactly generated R-linear ∞-category 𝒞 is dualizable. In this case, the dual is given by the Ind-completion 𝒞∨≔Ind(𝒞c,op), where 𝒞c denotes the subcategory of compact object and 𝒞c,op its opposite category. If 𝒞 is compactly generated, the evaluation functor ev𝒞 restricts along
𝒞c,op×𝒞c⊂𝒞∨×𝒞⟶𝒞∨⊗𝒞
to the restriction of the morphism object functor Mor𝒞(-,-) (see [35, Prop. D.7.2.3, Rem. D.7.7.6]). Under the assumption that 𝒞 is compactly generated, an R-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 LinCatRcpt−gen⊂LinCatRdual. We remark that this inclusion preserves both limits and colimits, as follows from the results of [17].
Remark 2.7.
The ∞-category LinCatRdual admits (small) colimits, and these are preserved by the forgetful functor LinCatRdual→LinCatR. This follows from combining [17, Prop. 1.65] and [34, Cor. 3.4.4.6].
The ∞-category LinCatRdual also admits (small) limits, described in [17]. In the setting of compactly generated ∞-categories, limits can be very concretely described: on the level of the underlying ∞-categories, the limit of a diagram D:Z→LinCatcpt−gen is computed by first restricting the values D(z) to compact objects, for all z∈Z, computing the limit of the resulting diagram in Cat∞, and then passing to the Ind-completion. This essentially follows from the fact that limits in the subcategory 𝒫rStL,cpt−gen⊂𝒫rL of compactly generated, stable ∞-categories and compact object-preserving functors are computed this way, since 𝒫rStL,cpt−gen≃Stidem is equivalent to the ∞-category of idempotent complete stable ∞-categories.
Definition 2.8.
Given a dualizable R-linear functor F:𝒞→𝒟 between dualizable R-linear ∞-categories with right adjoint G, we define the functor F∨:𝒞∨→𝒟∨ as the composite
is obtained by taking the opposite functor of the restriction f:𝒞c→𝒟c of F to compact objects and then applying Ind-completion.
Definition 2.9.
Given a dualizable R-linear ∞-category 𝒞, we denote by
𝒴:𝒞≃LinR(RModR,𝒞)→ev𝒞∘(id𝒞∨⊗(-))LinR(𝒞∨,RModR)
the R-linear Yoneda embedding. Its inverse is given by
LinR(𝒞∨,RModR)→((-)⊗id𝒞)∘coev𝒞LinR(RModR,𝒞)≃𝒞.
Lemma 2.10.
Let F:𝒞→𝒟 be a dualizable R-linear functor between dualizable R-linear ∞-categories with right adjoint G. Then the following diagram commutes.
The functor (-)∘G is left adjoint to (-)∘F, and the functor F∨ is hence dualizable.
Proof.
This readily follows from inspecting the definitions and using the triangle identities for the evaluation and coevaluation functors.∎
Lemma 2.11.
Let 𝒞 be a dualizable R-linear ∞-category. There exists an equivalence of R-linear functors RModR∨⊗𝒞∨⊗𝒞→RModR:
ev𝒞(C⊗Y,Z)≃evRModR(C,ev𝒞(Y,Z)).
Proof.
We note that RModR∨≃RModR and thus 𝒞∨⊗𝒞≃RModR∨⊗𝒞∨⊗𝒞. Composing with this equivalence, both functors in (2.2) yield ev𝒞(-,-), showing their equivalence.∎
Duals of bimodules
We again fix a base 𝔼∞-ring spectrum R. Suppose we are given two R-linear ring spectra A,A′. The ∞-category of A- A′-bimodules BModA′A(RModR) is equivalent to the ∞-category LinR(RModA,RModA′) of R-linear functors between the respective right module ∞-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-linear ∞-category. We are especially interested in the adjoints of functors 𝒞⊗𝒞∨→RModR or RModR→𝒞⊗𝒞∨. This corresponds as a special case to studying modules over the enveloping algebra Ae=A⊗RArev of some R-linear ring spectrum A. We have the following equivalences.
with 𝒴 the R-linear Yoneda embedding, maps an endofunctor X:𝒞→𝒞 to the functor ev𝒞∘(id𝒞∨⊗X). This shows that Θ𝒞−1 is essentially surjective. The triangle identity for ev𝒞 and coev𝒞 implies that Θ𝒞∘Θ𝒞−1≃idLinR(𝒞,𝒞). It follows that Θ𝒞−1 is faithful, and in fact a split inclusion on Hom spaces. Using that all objects Y∈LinR(𝒞∨⊗𝒞,RModR) are of the form Y≃ev𝒞∘(id𝒞∨⊗X), we find Θ𝒞−1∘Θ𝒞(Y)≃Y. Using that Θ𝒞 and Θ𝒞−1 are exact, it follows that Θ𝒞−1 is also full, showing that Θ𝒞−1 is an equivalence. Since Θ𝒞∘Θ𝒞−1≃idLinR(𝒞,𝒞), the inverse of Θ𝒞−1 is given by Θ𝒞.
For part (2), a similar argument as above applies, using that the equivalence of ∞-categories
LinR(𝒞,𝒞)≃LinR(RModR,LinR(𝒞,𝒞))≃LinR(RModR,𝒞⊗𝒞∨)
maps a functor X:𝒞→𝒞 to (X⊗id𝒞∨)∘coev𝒞.∎
Notation 2.13.
We denote by τ the R-linear equivalence 𝒞⊗𝒞∨≃𝒞∨⊗𝒞 which permutes the factors.
We can use the equivalences Θ𝒞 and Ξ𝒞 to define the dual of an R-linear endofunctor 𝒞→𝒞, considered as a functor 𝒞∨⊗𝒞→RModR or RModR→𝒞⊗𝒞∨.
Definition 2.14.
Let X∈LinR(𝒞,𝒞) be an R-linear endofunctor.
We call X left dualizable if Θ𝒞−1(X) admits an R-linear left adjoint. In this case, we call
X!≔Ξ𝒞(τ∘ladj(Θ𝒞−1(X)))∈LinR(𝒞,𝒞)
the left dual of X.
We call X right dualizable if Θ𝒞−1(X) admits an R-linear right adjoint. In this case, we call
X∗≔Ξ𝒞(τ∘radj(Θ𝒞−1(X)))∈LinR(𝒞,𝒞)
the right dual of X.
Proposition 2.15.
Let X∈LinR(𝒞,𝒞).
Xis left dualizable if and only if Ξ𝒞−1(X)admits a right adjoint, and in this case,
X!≃Θ𝒞(τ∘radj(Ξ𝒞−1(X))).
Xis right dualizable if and only if Ξ𝒞−1(X)admits a left adjoint, and in this case,
X∗≃Θ𝒞(τ∘ladj(Ξ𝒞−1(X))).
Proof.
The lemma follows from Lemma 2.16 and the observation that for any dualizable R-linear functor F:𝒜→ℬ, we have
LinR(radj(F),RModR)⊣LinR(F,RModR).∎
Lemma 2.16.
Let X∈LinR(𝒞,𝒞). There are commutative diagrams
and
with 𝒴 the R-linear Yoneda embedding (see Definition 2.9).
Proof.
The evaluation functor
ev𝒞⊗𝒞∨:𝒞∨⊗𝒞⊗𝒞⊗𝒞∨≃(𝒞⊗𝒞∨)∨⊗𝒞⊗𝒞∨⟶RModR
is, after reordering the factors of the tensor product, given by the tensor product of the evaluation functors of 𝒞 and 𝒞∨. Note that these two evaluation functors are themselves equivalent up to composition with τ:𝒞⊗𝒞∨≃𝒞∨⊗𝒞. The Yoneda embedding
LinR(RModR,𝒞⊗𝒞∨)≃𝒞⊗𝒞∨→𝒴LinR(𝒞∨⊗𝒞,RModR)
is thus given by the functor
(ev𝒞⊗ev𝒞)∘(id𝒞∨⊗(-)⊗id𝒞).
Using this, the commutativity directly follows from the triangle identities for the evaluation and coevaluation functors when inserting the descriptions of Θ𝒞−1(X),Ξ𝒞−1(X) in Lemma 2.12.∎
Remark 2.17.
We denote by LinRld(𝒞,𝒞)⊂LinR(𝒞,𝒞) the stable subcategories of left dualizable functors. We similarly denote by LinRrd(𝒞,𝒞)⊂LinR(𝒞,𝒞) the stable subcategory of right dualizable functors. Since passing to adjoints is functorial (see [31, Prop. 5.2.6.2]), there are exact functors
(-)!:LinRld(𝒞,𝒞)→LinR(𝒞,𝒞)op
and
(-)∗:LinRrd(𝒞,𝒞)→LinR(𝒞,𝒞)op.
More concretely, we find that LinRld(𝒞,𝒞)=LinR(𝒞,𝒞)c is given by the subcategory of compact objects. This follows from the observation that a compact object in
𝒞⊗𝒞∨≃𝒴LinR(𝒞∨⊗𝒞,RModR)≃LinR(𝒞,𝒞)
gives via the Yoneda embedding rise to an exact functor 𝒞∨⊗𝒞→RModR which also preserves filtered limits, and hence all limits, and thus admits a left adjoint by the adjoint functor theorem. If 𝒞 is compactly generated, then an endofunctor is right dualizable if and only if its image under Θ𝒞−1 in LinR(𝒞∨⊗𝒞,RModR) preserves compact objects, since in this case the right adjoint preserves colimits and is thus R-linear.
Lemma 2.18.
Let F:𝒞→𝒟 be a morphism in LinCatRdual with R-linear right adjoint G.
There exists a commutative diagram.
In particular, it follows that theR-linear functor
F!≔F∘(-)∘G:LinR(𝒞,𝒞)→LinR(𝒟,𝒟)
preserves compact objects, meaning left dualizable functors.
There exists a commutative diagram.
In particular, it follows that theR-linear functor
F∗≔G∘(-)∘F:LinR(𝒟,𝒟)→LinR(𝒞,𝒞)
preserves right dualizable functors.
Proof.
We only prove part (1); part (2) is analogous. It follows from
Smooth and proper linear <inline-formula><alternatives><mml:math display="inline"><mml:mi mathvariant="normal">∞</mml:mi></mml:math><tex-math><![CDATA[\(\infty\)]]></tex-math></alternatives></inline-formula>-categories
We fix an 𝔼∞-ring spectrum R and a dualizable R-linear ∞-category 𝒞.
Definition 2.19.
Let 𝒞 be as above.
The ∞-category 𝒞 is called smooth if id𝒞∈LinR(𝒞,𝒞) is left dualizable. In this case, the left dual id𝒞! is also called the inverse dualizing bimodule.
The ∞-category 𝒞 is called proper if the functor id𝒞 is right dualizable.
Note that we do not require smooth or proper ∞-categories to be compactly generated. However, if 𝒞 is compactly generated, then 𝒞 being proper is equivalent to the assertion that for any two compact objects X,Y∈𝒞c, the R-linear morphism object Mor𝒞(X,Y)∈RModR is compact.
We denote (-)∗=MorRModR(-,R):(RModRc)op→RModRc.
Definition 2.20.
Suppose that 𝒞 is compactly generated and proper. We call an R-linear endofunctor U∈LinR(𝒞,𝒞) a Serre functor of 𝒞 if there exists a natural equivalence
Mor𝒞(-1,-2)≃Mor𝒞(-2,U(-1))∗:𝒞c,op×𝒞c→RModRc.
Lemma 2.21.
Suppose that 𝒞 is compactly generated and proper, and let U,U′ be two Serre functors of 𝒞. Then U≃U′∈LinR(𝒞,𝒞).
Proof.
Since U and U′ are both Serre functors, there exist natural equivalences
Mor𝒞(-1,U(-2))≃Mor𝒞(-2,-1)∗≃Mor𝒞(-1,U′(-2)).
Applying MapRModR(R,-) to this equivalence yields
Map𝒞c(-1,U(-2))≃Map𝒞c(-1,U′(-2)).
It follows that U≃U′ on 𝒞c by (a corollary of) the Yoneda lemma (see, for instance, [11, Cor. 5.8.14]). Passing to Ind-completions shows U≃U′.∎
Lemma 2.22.
The following statements hold:
If𝒞is proper and compactly generated, the right dualid𝒞∗is a Serre functor of𝒞.
If𝒞is smooth and proper, the functorsid𝒞∗andid𝒞!are inverse equivalences.
Remark 2.23.
Part (1) of Lemma 2.22 is stated without proof in [35, Rem. 11.1.5.2].
Proof of Lemma <xref rid="j_JNCG663_stat_002_037" ref-type="statement">2.22</xref>.
We begin by proving part (1). We denote by
Mor^𝒞(-,-):𝒞∨×𝒞→RModR
the functor obtained by passing to Ind-completions from the restriction of Mor𝒞(-,-) to 𝒞c,op×𝒞c. Let X∈𝒞c and consider the adjunction
Using that Mor^𝒞(X,-)≃Mor^𝒞∨(-,X) and the fully faithfulness of the R-linear Yoneda embedding of 𝒞∨, we find that the right adjoint is given by evaluation at X, that is,
Informally, the left adjoint ladj(evX′) is given by
(c′↦F(c′))↦((c⊗c′)↦Mor^𝒞(X,c)⊗F(c′)).
The right adjoint of the evaluation functor ev𝒞:𝒞∨⊗𝒞→RModR is equivalent to (id𝒞∨⊗id𝒞∗)∘coev𝒞∨. Using the description of the Yoneda embedding in (2.3), it follows that the right adjoint of the functor
is given by (-)⊗Mor^𝒞(-,id𝒞∗(X)). Using the above adjunctions, the fully faithfulness of the R-linear Yoneda embedding and Lemma 2.11, we find the following equivalences in RModR, functorial in (X,Y)∈𝒞c×𝒞c,op:
Combining the above equivalences yields id𝒞∗∘id𝒞!≃id𝒞. The identity id𝒞!∘id𝒞∗≃id𝒞 arises from a similar argument by passing to the left adjoint of (2.4).∎
Definition 2.24.
Given a compactly generated R-linear ∞-category 𝒞, we denote by 𝒞fin⊂𝒞 the full subcategory of objects Y, satisfying that Mor𝒞(X,Y)∈RModR is compact for all X∈𝒞c. We also refer to the objects of 𝒞fin as finite.
The following lemma provides the analogue of part (1) of Lemma 2.22 for smooth, but not necessarily proper, R-linear ∞-categories.
Lemma 2.25.
Let 𝒞 be a compactly generated and smooth R-linear ∞-category. Then
Mor𝒞(X,Y)∗≃Mor𝒞(id!(Y),X),
functorial in X∈𝒞c and Y∈(𝒞fin)op.
In particular, this shows that if id𝒞! is an equivalence, then 𝒞fin⊂𝒞c.
Proof.
The exact inclusion 𝒞fin⊂𝒞 gives rise to an R-linear functor Ind𝒞fin→𝒞. The R-linear functor
ev𝒞fin:𝒞∨⊗Ind𝒞fin⟶𝒞∨⊗𝒞→ev𝒞RModR
preserves compact objects by the definition of 𝒞fin and thus admits an R-linear right adjoint radj(ev𝒞fin). We define the R-linear functor U:𝒞→Ind𝒞fin as the composite
which describes the composite of id𝒞! with the R-linear functor Ind𝒞fin→𝒞.
The proof of part (1) of Lemma 2.22 adapts with minimal changes to this setting and shows that
Mor𝒞(X,Y)∗≃Mor𝒞(Y,U(X)),
functorial in Y∈𝒞fin,op and X∈𝒞c. By the above adjunction, we have
Mor𝒞(Y,U(X))≃Mor𝒞(id𝒞!(Y),X),
which combined with (2.5) yields the desired equivalence.∎
Traces
We fix an 𝔼∞-ring spectrum R.
Definition 2.26.
Let 𝒞 be a dualizable R-linear ∞-category. Let E:𝒞→𝒞 be an R-linear endomorphism. The trace Tr(E) of E is defined as the R-linear endomorphism
RModR→coev𝒞𝒞⊗𝒞∨→E⊗id𝒞∨𝒞⊗𝒞∨→𝜏𝒞∨⊗𝒞→ev𝒞RModR.
In the following, we recall the construction of Hoyois–Scherotzke–Sibilla [23] of the S1-equivariant (∞,1)-categorical trace functor. This functor will also give rise to the Hochschild homology functor.
As a model for (∞,2)-categories, we use Barwick’s complete 2-fold Segal spaces. We let 𝑪 be a symmetric monoidal (∞,2)-category. We will primarily be interested in the case where 𝑪=LinCatR is the (∞,2)-category of R-linear stable and presentable ∞-categories (see [23, § 4.4]), where the notation PrL(RModR) is used. We have an associated complete (1-fold) Segal space (which models an (∞,1)-category) of symmetric monoidal oplax transfors [25]
End(𝑪)=Fun⊗oplax(Frrig(Bℕ),𝑪),
where Frrig(Bℕ) denotes the free rigid
By rigid, we mean that all objects are dualizable.
(∞,2)-category generated by Bℕ (see also [23, Def. 2.2]). The objects and morphisms in End(𝑪) can be concretely described as follows:
Objects are given by pairs (𝒞,E) with 𝒞∈𝑪 dualizable and E:𝒞→𝒞 an endomorphism in 𝑪.
A morphism (F,α):(𝒞,E)→(𝒟,E′) corresponds to a right dualizable morphism F:𝒞→𝒟 in 𝑪 together with an oplax-commutative square
meaning a 2-morphism α:FE⇒E′F in 𝑪.
We also consider the complete Segal space Ω𝑪 of endomorphisms of the monoidal unit of 𝑪. The trace is defined by [23, Defs. 2.9 and 2.11] as a symmetric monoidal functor between complete Segal spaces
Tr:End(𝑪)→Ω𝑪,
which is natural in 𝑪. Given a morphism (F,α):(𝒞,E)→(𝒟,E′) in End(LinCatR), the morphism
Tr(F,α):Tr(E)→Tr(E′)
in MapLinCatR(RModR,RModR) can be identified with the composition of the following natural transformations.
Using the triangle identities, the natural transformation ν above is defined as the composite of
where G denotes the right adjoint of F and cu the counit. The natural transformation ε is defined similarly. Further, Tr(F,α) is also equivalent to the composite
where u denotes the unit and β is the apparent morphism (see [3, Lem. 4.1]).
Finally, we turn to the S1-functoriality of the trace functor. For this, consider the complete Segal space Aut(𝑪)=Fun⊗oplax(Frrig(Bℤ),𝑪), which comes with a symmetric monoidal inclusion Aut(𝑪)⊂End(𝑪). The objects of Aut(𝑪) are given as pairs (𝒞,E) with 𝒞∈𝑪 dualizable and E:𝒞→𝒞 an equivalence. The self-action of the circle group S1=Bℤ induces an action on Aut(𝑪), which in turn induces an action on the space Map(Aut(𝑪),Ω(𝑪)) of functors, natural in 𝑪. The trace can be exhibited as a homotopy fixed point of the S1-action on Map(Aut(𝑪),Ω(𝑪)) (see [23, Thm. 2.14]). This S1-invariance datum results in an S1-action on the trace of any pair (𝒞,id𝒞)∈End(𝑪) as well as an S1-equivariant map Tr(F,idF) for any morphism (F,idF):(𝒞,id𝒞)→(𝒟,id𝒟) in End(𝑪).
Remark 2.27.
The datum of the S1-invariance of the trace functor is natural in the symmetric monoidal (∞,2)-category 𝑪. Thus, given a symmetric monoidal functor F:𝑪→𝑫 and a dualizable object 𝒞∈𝑪, we have an S1-equivariant equivalence F(Tr(𝒞,id𝒞))≃Tr(F(𝒞),idF(𝒞)).
We can specialize this to the symmetric monoidal functor ∗→𝑪 from the terminal symmetric monoidal (∞,2)-category, mapping the unique object of ∗ to the unit object 1C of 𝑪. Since the S1-action on the trace Tr(∗,id∗) is necessarily trivial, this shows that the S1-action on Tr(1C,id1C) is also trivial. In particular, this holds for 𝑪=LinCatR with 1C=RModR and Tr(RModR,idRModR)≃R.
Hochschild homology
We fix an 𝔼∞-ring spectrum R. Given a dualizable R-linear ∞-category 𝒞, its R-linear Hochschild homology is defined as the value of the trace at R:
HH(𝒞)≔ev𝒞∘τ∘coev𝒞(R)=Tr(id𝒞)(R)∈RModR.
Note that Tr(id𝒞) is an R-linear functor and thus fully determined by its value HH(𝒞). When R is the sphere spectrum, HH(𝒞) is also called topological Hochschild homology. When R=k is a commutative ring, HH(𝒞) describes the usual k-linear Hochschild homology.
The dual Hochschild homology is defined as
HH(𝒞)∗≔MorRModR(HH(𝒞),R).
The fixed points (limit over BS1) of the S1-action on HH(𝒞) are denoted by HH(𝒞)S1. We will refer to HH(𝒞)S1 as the negative cyclic homology of 𝒞, as HH(𝒞)S1 recovers the usual negative cyclic homology when R=k is a commutative ring. The orbits (colimit over BS1) of the S1-action on HH(𝒞) are denoted HH(𝒞)S1, and we similarly call HH(𝒞)S1 the cyclic homology. The dual cyclic homology is given by HH(𝒞)S1∗=MorRModR(HH(𝒞)S1,R). There are natural maps HH(𝒞)S1→HH(𝒞) and HH(𝒞)S1∗→HH(𝒞)∗.
Notation 2.28.
Let F:𝒞→𝒟 be a morphism in LinCatRdual. We denote by
HH(F):HH(𝒞)→HH(𝒟) the evaluation at R of Tr(F,idF). We further define HH(𝒟,𝒞)=cofHH(F).
HH(F)S1:HH(𝒞)S1→HH(𝒟)S1 the induced map. In the same way, we define HH(𝒟,𝒞)S1=cof(HH(F)S1).
Let F:𝒟→𝒞 be a morphism in LinCatRdual. We denote by
HH(F)∗:HH(𝒞)∗→HH(𝒟)∗ the dual map obtained by precomposition with HH(F). We further define HH(𝒟,𝒞)∗=cofHH(F)∗.
HH(F)S1∗:HH(𝒞)S1∗→HH(𝒟)S1∗ the induced map. In the same way, we
define HH(𝒟,𝒞)S1∗=cof(HH(F)S1∗).
The notation in (1) and (2) will usually be used when discussing left Calabi–Yau structures on F, whereas the notation in (3) and (4) will usually be used when discussing right Calabi–Yau structures on F. This is why we swap the roles of 𝒞 (the ‘Calabi–Yau boundary’) and 𝒟 (the relative Calabi–Yau category) as domain and target.
Lemma 2.29.
Let 𝒞 be a dualizable R-linear ∞-category, and let E:𝒞→𝒞 be R-linear.
If𝒞is smooth, thenTr(E)(R)is canonically equivalent to
MorLinR(𝒞,𝒞)(id𝒞!,E).
In particular, we haveHH(𝒞)≃MorLinR(𝒞,𝒞)(id𝒞!,id𝒞).
If𝒞is proper, then
Tr(E)(R)∗≔MorRModR(Tr(E)(R),R)
is canonically equivalent to
MorLinR(𝒞,𝒞)(E,id𝒞∗).
In particular, we haveHH(𝒞)∗≃MorLinR(𝒞,𝒞)(id𝒞,id𝒞∗).
Proof.
Suppose that 𝒞 is smooth. Then we have an adjunction
Suppose the dualizable R-linear ∞-category 𝒞 is smooth. If we make two different choices of left duals/adjoints
id𝒞!=Ξ𝒞(ladj(ev𝒞)),(id𝒞!)′=Ξ𝒞(ladj(ev𝒞)′)
and two choices of units, there is a contractible space of equivalences α:(id𝒞!)′≃id𝒞!, compatible with the unit (see [11, Prop. 6.1.9]). Any such equivalence α assembles with the equivalences from Lemma 2.29 into a commutative diagram as follows.
Stated differently, this means that the equivalence in part (1) of Lemma 2.29 is independent of the choice of left dual. A similar statement holds for the equivalence in part (2).
Construction 2.31.
Let 𝒞,𝒟 be dualizable R-linear ∞-categories.
Case 1.
Suppose that 𝒞,𝒟 are smooth.
Let F:𝒞→𝒟 be a morphism in LinCatRdual and G the R-linear right adjoint of F. We denote by
F!(-)=F∘(-)∘G:LinR(𝒞,𝒞)⟶LinR(𝒟,𝒟)
the functor from Lemma 2.18 and by cu:F!(id𝒞)→id𝒟 the counit transformation of F⊣G. We define the unitu~:id𝒟!→F!(id𝒞!) as the image under Ξ𝒟 of the natural transformation
from Lemma 2.18. The transformation u~ is indeed a unit if F admits a left adjoint (see Lemma 2.34).
Case 2.
Suppose that 𝒞,𝒟 are proper.
Let F:𝒟→𝒞 be a morphism in LinCatRdual and G the R-linear right adjoint of F. Consider the functor
F∗(-)=G∘(-)∘F:LinR(𝒞,𝒞)⟶LinR(𝒟,𝒟)
from Lemma 2.18 and denote by u:id𝒟→F∗(id𝒞) the unit of F⊣G. Let E∨ denote the right adjoint of F∨. Applying Ξ𝒞∨−1 to the counit F∨E∨→id𝒞 defines a natural transformation
(F∨⊗F)∘coev𝒟∨⟶coev𝒞∨.
We use this to define the counitcu~:F∗(id𝒞∗)→id𝒟∗ as the image under Θ𝒟 of the natural transformation
composed with the identification Θ𝒟(ladj(coev𝒞∨)∘(F∨⊗F))≃F∗(id𝒞∗) as stated in Lemma 2.18.
The following proposition describes the Hochschild homology functor in terms of the corresponding morphisms between bimodules.
Proposition 2.32
([3, Prop. 4.4]). Let F:𝒞→𝒟 be a map in LinCatRdual with 𝒞 smooth. Consider a morphism ξ:R[i]→Tr(id𝒞), corresponding via Lemma 2.29 to a natural transformation α:id𝒞![i]→id𝒞.
Suppose that𝒟is smooth. Then the morphism (2.7) can be identified with
cu∘F!(α)∘u~∈ExtLinR(𝒟,𝒟)−i(id𝒟!,id𝒟).
Proof.
The proof of part (1) can be found in [3, Prop. 4.4]; we also spell out its dual version in the proof of Proposition 2.33. Part (2) is stated in loc. cit.; we spell out the proof for the convenience of the reader. Using part (1), it suffices to show that the composite
Applying the triangle identity of the adjunction F⊣G, we find that this agrees with the above.∎
Given (𝒞,E)∈End(LinCatR), we denote by Tr(𝒞,E)∗ the composite of Tr(𝒞,E) and MorRModR(-,R):RModR→RModR. The analogue of Proposition 2.32 for the dual Hochschild homology is as follows.
Proposition 2.33.
Let F:𝒟→𝒞 be a map in LinCatRdual with 𝒟 proper. Consider a morphism ξ:R[i]→Tr(id𝒞)∗, corresponding via Lemma 2.29 to a natural transformation α:id𝒞→id𝒞∗[−i].
Suppose that𝒞is proper. Then the morphism (2.8) can be identified with
cu~∘F∗(α)∘u∈ExtLinR(𝒟,𝒟)−i(id𝒟,id𝒟∗).
Proof.
The proof of part (1) is dual to the proof of part (1) of Proposition 2.32. Using Lemma 2.29, the identity on id𝒞∗ gives rise to a morphism μ:R[i]→Tr(id𝒞∗)[i]. The statement now follows from the following commutative diagram.
Note that the commutativity of the rightmost triangle follows from dualizing equation (2.6).
Part (2) can be shown as in the proof of Proposition 2.32.∎
Lemma 2.34.
Let 𝒞,𝒟∈LinCatRdual.
Suppose that𝒞,𝒟are smooth. LetF:𝒞→𝒟be a map inLinCatRdualwhich admits a left adjointE. Then there exists a natural equivalenceF!(id𝒞!)≃FEid𝒟!such that the composite of
u~:id𝒟!→F!(id𝒞!)
with this equivalence describes a unit ofE⊣Fcomposed withid𝒟!.
Suppose that𝒞,𝒟are proper. LetF:𝒟→𝒞be a map inLinCatRdualwhich admits a left adjointE. Then there exists a natural equivalenceF∗(id𝒞∗)≃id𝒟∗EFsuch that the composite of
cu~:F∗(id𝒞∗)→id𝒟∗
with this equivalence describes a counit ofE⊣Fcomposed withid𝒟∗.
Proof.
We only prove part (1); part (2) is similar. We have
ev𝒞∘(E∨⊗id𝒞)≃ev𝒟∘(id𝒟∨⊗F).
The units of the adjunctions
(F∨⊗id𝒞)∘ladj(ev𝒞)⊣ev𝒞∘(E∨⊗id𝒞)
and
(id𝒟∨⊗E)∘ladj(ev𝒟)⊣ev𝒟∘(id𝒟∨⊗F)
are therefore equivalent. This gives rise to the following commutative diagram.
Consequently, the naturality of the unit id𝒟∨⊗id𝒟→id𝒟∨⊗FE and the counit id𝒟∨⊗ladj(ev𝒟)ev𝒟→id𝒟∨⊗id𝒟 gives rise to the following commutative diagram.
Postcomposing the upper diagram with ladj(ev𝒟) and combining it with the lower diagram, we see that the definition of the natural transformation u~:id𝒟!→F!(id𝒞!) in Construction 2.31 is equivalent to the image under Ξ𝒟 of the natural transformation
The desired description of u~ now follows via the triangle identity for the adjunction ladj(ev𝒟)⊣ev𝒟.∎
The following lemma shows that the natural transformations u~ and cu~ can also be seen as the adjoints of the counit and unit of F⊣G.
Lemma 2.35.
The following statements hold:
LetF:𝒞→𝒟be a morphism inLinCatRdualwith𝒞,𝒟smooth. Denote the right adjoint ofFbyG. The image underΞ𝒞−1of the unitu~:id𝒟!→Fid𝒞!Gis left adjoint to the counitev𝒟∘(id𝒟∨⊗FG)→ev𝒟.
LetF:𝒟→𝒞be a morphism inLinCatRdualwith𝒞,𝒟proper. Denote the right adjoint ofFbyG. The image underΘ𝒞−1of the counitcu~:Gid𝒞∗F→id𝒞∗is left adjoint to the unitev𝒟→ev𝒟∘(id𝒟∨⊗GF).
Proof.
We only prove part (1); part (2) can be proven similarly. Denote the right adjoint of F∨ by E∨. Upon passing to right adjoints, the natural transformation Ξ𝒞−1(u~):ladj(ev𝒟)→(F∨⊗F)∘ladj(ev𝒞) induces the following natural transformation:
where the third natural transformation uses the counits of F⊣G and F∨⊣E∨. This natural transformation fits into the following commutative diagram:
exhibiting it as the counit ev𝒟∘(id𝒟∨⊗FG)→ev𝒟 by further triangle identities.∎
Relative Calabi–Yau structures
The goal of this section is to introduce R-linear relative Calabi–Yau structures and describe essential features of their theory. We begin in Sections 3.1 and 3.2 with their definitions. After the short Section 3.3 on the behavior of Calabi–Yau structures under tensor products, we generalize the gluing properties of relative Calabi–Yau structures of [2] to the R-linear setting in Section 3.4. For the entire section, we fix an 𝔼∞-ring spectrum R.
Left Calabi–Yau structures
Let F:𝒞→𝒟 be a dualizable R-linear functor between dualizable and smooth R-linear ∞-categories. Part (2) of Proposition 2.32 shows that an R-linear relative Hochschild homology class σ:R[n]→HH(𝒟,𝒞) amounts to a diagram in LinR(𝒟,𝒟)
together with a choice of null-homotopy of the composite id𝒟!→id𝒟[1−n]. The null-homotopy allows us to extend the diagram to a diagram with horizontal fiber and cofiber sequences as follows.
We call the Hochschild homology class σnon-degenerate if all the vertical maps in diagram (3.1) are equivalences.
Definition 3.1.
Let F be as above.
A weak left n-Calabi–Yau structure on the functor F consists of a non-degenerate Hochschild homology class σ:R[n]→HH(𝒟,𝒞). If F=0, we also say that 𝒟 carries a weak left n-Calabi–Yau structure.
A left n-Calabi–Yau structure on the functor F is composed of a negative cyclic homology class η:R[n]→HHS1(𝒟,𝒞), the composition of which with HHS1(𝒟,𝒞)→HH(𝒟,𝒞) defines a non-degenerate Hochschild homology class. If F=0, we also say that 𝒟 carries a left n-Calabi–Yau structure.
Weak left n-Calabi–Yau structures are also sometimes called bimodule n-Calabi–Yau structures.
Remark 3.2.
The notion of weak left Calabi–Yau structure on a functor F only depends on the functor and the relative Hochschild class and not on any further choices made in its definition. This includes choices of adjoints and (co)units. For example, we make a choice of right adjoint of F together with the counit; the space of such choices is however contractible. Inspecting the definition, one finds that making a different choice yields an equivalent diagram in (3.1) and thus the same condition of the Hochschild class being non-degenerate.
Right Calabi–Yau structures
Let F:𝒟→𝒞 be a dualizable R-linear functor between dualizable and proper R-linear ∞-categories. Part (2) of Proposition 2.33 shows that an R-linear dual relative Hochschild homology class σ:R[n]→HH(𝒟,𝒞)∗ amounts to a diagram in LinR(𝒟,𝒟)
together with a choice of null-homotopy of the composite id𝒟→id𝒟∗[1−n]. We extend the diagram to a diagram with horizontal fiber and cofiber sequences as follows.
As in the smooth case, we call the dual Hochschild homology class σnon-degenerate if all the vertical maps in the above diagram are equivalences.
Definition 3.3.
Let F be as above.
A weak right n-Calabi–Yau structure on the functor F consists of a non-degenerate dual Hochschild homology class σ:R[n]→HH(𝒟,𝒞)∗. If 𝒞=0, we also say that 𝒟 carries a weak right n-Calabi–Yau structure.
A right n-Calabi–Yau structure on the functor F consists of a dual cyclic homology class η:R[n]→HH(𝒟,𝒞)S1∗, whose composition with HH(𝒟,𝒞)S1∗→HH(𝒟,𝒞)∗ defines a non-degenerate dual Hochschild homology class. If 𝒞=0, we also say that 𝒟 carries a right n-Calabi–Yau structure.
Remark 3.4.
Assume that 𝒟 is compactly generated. A weak right n-Calabi–Yau structure on 𝒟 equivalently consists of an equivalence in RModR
Mor𝒟(X,Y)≃Mor𝒟(Y,X[n])∗,
bifunctorial in X,Y∈𝒟c.
In good situations, a relative right n-Calabi–Yau structure on 𝒟 gives rise to a duality of a subfunctor of Mor𝒟(-,-), generalizing the equivalence (3.3) (see [8]).
Remark 3.5.
It is also possible to make sense of relative right Calabi–Yau structures on some non-proper k-linear ∞-categories, namely those arising as the derived ∞-category of a dg algebra whose homology is finite dimensional in each degree. We refer to the recent work [28] for this notion and the construction of many examples related to cluster categories.
Finally, we comment on the relation with dg categorical left and right Calabi–Yau structures.
Recall that the passage to derived ∞-categories defines a functor 𝒟(-):N(dgCatk)→LinCatk, with dgCatk the 1-category of k-linear dg categories.
Lemma 3.6.
Let f:A→B be a dg functor. Then F≔𝒟(f):𝒟(A)→𝒟(B) is a colimit-preserving functor between compactly generated k-linear ∞-categories.
Suppose thatAandBare smooth. Then𝒟(A),𝒟(B)are smooth as well. Further, any weak leftn-Calabi–Yau structure offinduces a weak leftn-Calabi–Yau structure ofFand vice versa.
Suppose thatAandBare proper. Then𝒟(A),𝒟(B)are proper as well. Further, any weak rightn-Calabi–Yau structure offinduces a weak rightn-Calabi–Yau structure ofFand vice versa.
Proof.
We may assume that A and B are cofibrant dg categories (with respect to the quasi-equivalence model structure). We first show that 𝒟(A⊗Aop)≃𝒟(A)⊗𝒟(Aop), where the former tensor product is of dg categories and the latter in LinCatk. A similar argument shows that 𝒟(B⊗Bop)≃𝒟(B)⊗𝒟(Bop). We can compute A⊗Aop as the colimit of the 2-sided bar construction, which is a diagram Δop→dgCatk, mapping [n] to A×dgModk×n×B. This diagram is cofibrant with respect to the Reedy model structure on Fun(Δop,dgCatk), meaning that all latching morphisms are cofibrations [22, Def. 15.3.3]. Thus, the colimit of the 2-sided bar construction computes its homotopy colimit (see also [22, Thm. 15.10.8]). The image under 𝒟(-) of the homotopy colimit can further be identified with the colimit of the ∞-categorical 2-sided bar construction [34, Constr. 4.4.2.7], computing 𝒟(A)⊗𝒟(Aop)∈LinCatk.
To the author’s knowledge, it has not been shown that there is a symmetric monoidal functor of (∞,2)-categories lifting 𝒟(-). However, the above shows that the ∞-functor 𝒟(-):N(dgCatk)→LinCatk preserves the tensor product of two cofibrant dg categories and thus maps the evaluation and coevaluation bimodules to the evaluation and coevaluation functors. The functor thus (non-coherently) identifies the two traces, and hence (non-coherently) the functoriality of Hochschild homology. To identify the S1-actions on the traces, a coherent identification of the traces would be necessary.
To obtain the statement from the lemma, all that remains to note is that the non-degeneracy conditions on relative (possibly dual) Hochschild classes of f and F coincide, which follows from Proposition 2.32 and [3, Prop. 4.4] in the smooth case and a similar argument in the proper case.∎
Behavior under tensor productsLemma 3.7.
Let 𝒞,𝒟 be dualizable R-linear ∞-categories. There is a canonical equivalence
HH(𝒞)⊗HH(𝒟)≃HH(𝒞⊗𝒟).
Proof.
Using that ev𝒞⊗𝒟≃ev𝒞⊗ev𝒟 and coev𝒞⊗𝒟≃coev𝒞⊗coev𝒟, we find
We note that the equivalence in Lemma 3.7 is S1-equivariant.
Remark 3.8.
If 𝒞,𝒟 are smooth, then 𝒞⊗𝒟 is also smooth, and we have id𝒞⊗𝒟!≃id𝒞!⊗id𝒟!. Similarly, if 𝒞,𝒟 are proper, then 𝒞⊗𝒟 is proper and id𝒞⊗𝒟∗≃id𝒞∗⊗id𝒟∗.
In the smooth case, a pair of morphisms α:id𝒞!→id𝒞[−n], β:id𝒟!→id𝒟[−m] gives, under the identifications from Lemmas 2.29 and 3.7, rise to the morphism
id𝒞⊗𝒟!≃id𝒞!⊗id𝒟!→α⊗βid𝒞[−n]⊗id𝒟[−m]≃id𝒞⊗𝒟[−n−m].
A similar assertion holds in the proper case.
The following proposition shows that the tensor product of a Calabi–Yau functor with a Calabi–Yau category is again Calabi–Yau. A similar statement is proven in [3, Prop. 6.4].
Proposition 3.9.
Let 𝒞,𝒟,ℰ be dualizable R-linear ∞-categories.
Assume that𝒞,𝒟,ℰare smooth. LetF:𝒞→𝒟be a dualizableR-linear functor, and letη:R[n]→HH(𝒟,𝒞)S1andη′:R[m]→HH(ℰ)S1be left Calabi–Yau-structures onFandℰ, respectively. The class
Assume that𝒞,𝒟,ℰare proper. LetF:𝒟→𝒞be a dualizableR-linear functor, and letη:R[n]→HH(𝒟,𝒞)S1∗andη′:R[m]→HH(ℰ)S1∗be right Calabi–Yau structures onFandℰ, respectively. The class
We only prove part (1); part (2) is analogous. The Hochschild homology class R[n+m]→HH(𝒟⊗ℰ,𝒞⊗ℰ)S1→HH(𝒟⊗ℰ,𝒞⊗ℰ) gives rise to the following diagram in LinR(𝒟⊗ℰ,𝒞⊗ℰ), up to equivalence.
The horizontal sequences in the above diagram are fiber and cofiber sequences as tensor products of such with idℰ! or idℰ[m]. The vertical maps are equivalences as tensor products of equivalences, showing the non-degeneracy of the Hochschild homology class.∎
Gluing Calabi–Yau structures
In this section, we discuss a generalization of the gluing theorem for left Calabi–Yau structures on k-linear dg categories (see [2, Thm. 6.1]) to left and right Calabi–Yau structures on dualizable R-linear ∞-categories (see Theorems 3.15 and 3.16). The gluing theorems boil down to a simple description of objects in pullbacks/pushouts of stable, presentable ∞-categories in terms of their restrictions given in Lemma 3.11.
Consider the simplicial set Z=Δ{0,1}∐Δ{0}Δ{0,1′} describing a span with objects 0,1,1′ and two non-degenerate 1-simplices 0→1,1′. We fix a diagram D:Z→LinCatR with colimit 𝒞, satisfying that D maps each morphism to a functor admitting an R-linear right adjoint. For z∈Z, we denote 𝒞z=D(z) and can depict the colimit diagram of D as follows.
Above iz:𝒞z→𝒞 denotes the functor from the colimit cone. We further denote by jz=radj(iz) the right adjoint, and cuz the counit of iz⊣jz, for z∈Z.
The fact that counits compose to counits provides us with a commutative square
ϕD:Z▷→LinR(𝒞,𝒞)
which can be depicted as follows.
Proposition 3.10.
The square ϕD:Z▷→LinR(𝒞,𝒞) is biCartesian.
Proof.
Using that the forgetful functor LinR(𝒞,𝒞)→Fun(𝒞,𝒞) reflects finite colimits (see Lemma 2.5) and that colimits in functor categories are computed pointwise (see [31, Cor. 5.1.2.3]), the statement reduces to Lemma 3.11.∎
Lemma 3.11.
Let c∈𝒞 and consider the diagram
ϕc:Z▷→𝒞,
given by evaluating ϕD at c. Then ϕc describes a biCartesian square in 𝒞.
Proof.
We can identify the colimit 𝒞 of D with the ∞-category of coCartesian sections of the Grothendieck construction π:Γ(radj(D))→Zop of the diagram obtained from D by passing to right adjoint functors, that is, the coCartesian fibration classified by radj(D) (see
[36, Prop. 05RX]). Denote by ℒ the ∞-category of all sections of π and by κ:𝒞↪ℒ the fully faithful inclusion with left adjoint ζ. For z∈Z, we denote by j~z:ℒ→𝒞z the evaluation functor at z, satisfying that j~z∘κ≃jz. The left adjoint of j~z is denoted i~z, which satisfies ζ∘i~z≃iz. An object c∈𝒞 corresponding to a coCartesian section of π is of the form
with ci∈𝒞i. By [31, Prop. 4.3.2.17], i~z is given by the π-relative left Kan extension functor, and the objects i~zjz(c), z∈Z are hence given as follows.
These assemble into a square ϕ~c in ℒ of the form
which restricts at 0∈Z to the constant diagram with value c0, up to equivalence, and at i=1,1′∈Z to the fiber sequence of the map ci→0 in 𝒞i. Using that limits in the ∞-category ℒ of sections of the Grothendieck construction are computed componentwise in Z, it follows that ϕ~c is a limit diagram in ℒ. Using that ζ:ℒ→𝒞 is exact, we conclude that ϕc≃ζ∘ϕ~c is a limit diagram as well.∎
Proposition 3.10 implies that R-linear smoothness is preserved under finite colimits along dualizable functors. A variant of this observation for k-linear ∞-categories appears in [40, Lem. 8.21].
Corollary 3.12.
Let W be a finite simplicial set and D:W→LinCatRdual a functor taking values in smooth R-linear ∞-categories. Then the colimit 𝒞 of D in LinCatRdual is also smooth.
Proof.
Any finite colimit can be computed in terms of pushouts and finite coproducts. Smoothness is clearly preserved under finite coproducts. It thus suffices to check that the pushout of a span of compactly generated, smooth R-linear ∞-categories along compact object-preserving functors is again smooth. This follows from combining the fact that the forgetful functor LinCatRdual→LinCatR preserves colimits (see [17, Prop. 1.65]), with Lemma 3.11, part (1) of Lemma 2.18 and the fact that pushouts of compact objects are again compact.∎
We next discuss a dual version of Proposition 3.10 for limit diagrams of dualizable ∞-categories. Fix a cospan D:Zop→LinCatRdual. For z∈Zop, we denote 𝒞z=D(z). Let D~:(Zop)◁→LinCatRdual be a cone over D, with tip denoted 𝒞. We depict this cone as follows.
We denote by kz the right adjoint of jz for z∈Zop. The units of these adjunctions assemble into the diagram ϕD:(Zop)◁→LinR(𝒞,𝒞), depicted as follows.
Proposition 3.13.
If D~ is a limit cone in LinCatRdual, then ϕD is a biCartesian square.
Proof.
We show in the following that if the composition of D~ with the forgetful functor LinCatRdual→LinCatR defines a limit cone, then ϕD is a biCartesian square. Note that the pullback 𝒞1 of D in LinCatRdual embeds fully faithfully into the pullback 𝒞2 of D in LinCatR via an R-linear functor ι:𝒞1↪𝒞2 admitting an R-linear right adjoint (see [17, Prop. 1.87]). Hence the square ϕD in LinR(𝒞1,𝒞1) arises from applying to the biCartesian square LinR(𝒞2,𝒞2) the exact functor radj(ι)∘(-)∘ι, which shows that ϕD is biCartesian.
We thus suppose that D~ expresses 𝒞 as the pullback of D in LinCatR. Using Lemma 2.5 and that colimits in functor categories are computed pointwise (see [31, Cor. 5.1.2.3]), it suffices to show that for any c∈𝒞, the diagram ϕc≔ϕD(c) is biCartesian in 𝒟. By passing to opposite ∞-categories (which exchanges left and right adjoints, as well as unit and counit maps), the argument from the proof of Lemma 3.11 directly applies to show that ϕc is biCartesian.∎
Corollary 3.14.
Let W be a finite simplicial set and D:W→LinCatRdual a functor taking values in proper R-linear ∞-categories. Then the limit 𝒞 of D in LinCatRdual is also proper.
Proof.
Since finite limits are generated by products (for which the statement is clear) and pullbacks, the statement follows from combining Proposition 3.13 and part (2) of Lemma 2.18.∎
The above discussion provides us with the tools needed for proving the gluing results for Calabi–Yau structures. We begin with the gluing of left Calabi–Yau structures. For this, fix a colimit diagram in LinCatRdual valued in smooth ∞-categories of the following form.
We form the following diagram in RModR.
The outer square of the above diagram, though not necessarily biCartesian, induces a morphism X→HH(𝒟,ℬ1⊕ℬ3)S1[−1]. A class R[n]→X[1] corresponds to two classes R[n]→HH(𝒞1,ℬ1×ℬ2)S1,HH(𝒞2,ℬ2×ℬ3)S1, whose restrictions to HH(B2)S1[1] are not identical but differ exactly by a reversal of the sign, that is, composition with −HH(idℬ2)S1. In this case, we say that the restrictions of the classes to HH(B2)S1[1] are compatible.
Theorem 3.15.
Consider two classes ηi:R[n]→HH(𝒞i,ℬi×ℬi+1)S1, with i=1,2, whose restrictions to HH(ℬ2)S1[1] are compatible, and let η:R[n]→HH(𝒟,ℬ1×ℬ3)S1 be the arising class. If η1 and η2 define left n-Calabi–Yau structures on the functors
ℬi×ℬi+1⟶𝒞i,i=1,2,
then η defines a left n-Calabi–Yau structure on the functor
ℬ1×ℬ3⟶𝒟.
Proof.
For 𝒳=ℬ1,ℬ2,ℬ3,𝒞1,𝒞2, denote by i𝒳:𝒳→𝒟 the functor from (3.5). Let j𝒳 be the right adjoint of i𝒳. Since the restriction of (3.5) to ℬ2,𝒞1,𝒞2 and 𝒟 is a pushout diagram, we find by Proposition 3.10 a biCartesian square in LinR(𝒟,𝒟), which is depicted as follows.
The sequence
iℬ2jℬ2→(α1,−α2)i𝒞1j𝒞1⊕i𝒞2j𝒞2→(β1,β2)id𝒟
is hence a fiber and cofiber sequence.
Using the pasting law for biCartesian squares, this gives rise to the following commutative diagram in LinR(𝒟,𝒟), where all squares are biCartesian and all objects are compact.
The image under (-)! of the counit cu𝒳 is given by the unit u~:id𝒟!→i𝒳id𝒳!j𝒳 (see Lemma 2.35). Applying the exact contravariant functor (-)! to (3.6) yields the following diagram, up to equivalence.
The classes σ1,σ2 define an equivalence between the lower left squares and upper right squares of the lower diagram and the (1−n)-th suspension of the upper diagram. These equivalences extend to an equivalence of the entire diagrams by using that the lower right and upper left squares are biCartesian. Restricting the equivalence to the outer biCartesian squares provides us with a diagram in LinR(𝒟,𝒟)
with horizontal fiber and cofiber sequences. This diagram arises from the class
R[n]→𝜂HH(𝒟,ℬ1×ℬ3)S1→HH(𝒟,ℬ1×ℬ3),
thus showing that η defines a left n-Calabi–Yau structure.∎
We next describe the gluing properties of right Calabi–Yau structures along pullbacks. For this, we consider a limit diagram in LinCatRdual valued in proper R-linear ∞-categories of the following form.
We form the following diagram in RModR.
Similar to the smooth case, a class in X consists of classes in HH(𝒞i,ℬi×ℬi+1)S1∗[−1], with i=1,2, whose restrictions to HH(ℬ2)S1∗ differ by sign, and we again call such classes compatible.
Theorem 3.16.
Consider two classes ηi:R[n]→HH(𝒞i,ℬi×ℬi+1)S1∗, with i=1,2, whose restrictions to HH(ℬ2)S1∗[1] are compatible, and let η:R[n]→HH(𝒟,ℬ1×ℬ3)S1∗ be the arising class. If η1 and η2 define right n-Calabi–Yau structures on the functors
𝒞i⟶ℬi×ℬi+1,i=1,2,
then η also defines a right n-Calabi–Yau structure on the functor
𝒟⟶ℬ1×ℬ3.
Proof.
Using Proposition 3.13, this can be proven in the same way as Theorem 3.15.∎
Remark 3.17.
The analogues of Theorems 3.15 and 3.16 for weak relative Calabi–Yau structures clearly hold as well.
Perverse schobers on surfaces with boundarySurfaces, ribbon graphs and line fieldsDefinition 4.1.
By a surface, we will mean a smooth, connected, oriented surface S with non-empty boundary ∂S and interior S∘. We will also assume that S is compact, unless stated otherwise. Note that if S is compact, the boundary ∂S consists of a disjoint union of circles.
A marked surface (S,M) consists of a surface and a non-empty finite set M⊂∂S of marked points, lying on the boundary of S. We do not require that each connected component of ∂S contains at least one marked point.
Definition 4.2.
We define as follows:
A graph G consists of two finite sets G0 of vertices and HG of halfedges (mostly simply denoted by H) together with an involution τ:H→H and a map σ:H→G0.
Let G be a graph. An edge of G is defined to be an orbit of τ. The set of edges is denoted by G1. An edge is called internal if the orbit contains two elements and called external if the orbit contains a single element. An internal edge is called a loop at a vertex v∈G0 if it consists of two halfedges, both being mapped under σ to v. We denote the set of external edges of G by G1∂.
A ribbon graph consists of a graph G together with a choice of a cyclic order on the set H(v) of halfedges incident to each vertex v.
We will always assume graphs to be connected.
Definition 4.3.
Let G be a graph. We define the exit path category Exit(G) of G to be the nerve of the 1-category with
objects the vertices and edges of G and
a non-identity morphism of the form v→e for every vertex v and incident edge e. If e is a loop at v, then there are two morphisms v→e.
The geometric realization |G| of G is defined as the geometric realization |Exit(G)| of Exit(G).
Remark 4.4.
A graph G whose geometric realization |G| is embedded into an oriented surface S inherits a canonical ribbon graph structure by requiring the halfedges at any vertex to be ordered in the counterclockwise direction.
Definition 4.5.
Let S be a marked surface. A spanning graph for S consists of a graph G together with an embedding i:|G|⊂S\M satisfying that
i is a homotopy equivalence,
i maps ∂|G| to ∂S, and
the restriction ∂|G|→∂S\M gives a homotopy equivalence with the boundary components which do not contain marked points.
We consider a spanning graph of S as endowed with the canonical ribbon graph structure arising from the embedding into S.
We now turn to line fields and framings on surfaces.
Definition 4.6.
Let Σ be a possibly non-compact surface.
A line field ν on Σ is a section of the projectivized tangent bundle ℙTΣ.
Assume that Σ is equipped with a line field ν, and let γ:S1≃[0,1]/(0∼1)→Σ be a smooth, immersed loop. We denote by W(γ)∈ℤ≃π1ℙTγ(0)Σ the winding number of γ with respect to ν.
For the equivalence ℤ≃π1ℙTγ(0)Σ, we use the convention that 1 corresponds to a counterclockwise half-turn.
We let Fr(Σ)→Σ denote the principal Gl(2,ℝ)-bundle of frames (i.e., ordered bases) of the tangent bundle TΣ.
A framing ξ on Σ is a section of (TΣ\{0})/ℝ+.
The winding number W(γ) of an immersed loop γ can be obtained as follows: one chooses any homotopy in ℙTγ(0)Σ from the tangent vector γ˙(0) to ν(γ(0)), extends this homotopy to a homotopy of sections Γ([0,1];γ∗ℙTΣ) from γ˙ to γ∗ν and then composes the two homotopies between γ˙(1)=γ˙(0) and ν(γ(0))=ν(γ(1)) to obtain a loop in ℙTγ(0)Σ. Informally, this counts the number of half-rotations of the tangent field along γ with respect to the line field.
Remark 4.7.
By projecting onto the first element of the ordered basis, we obtain a map of fiber bundles Fr(Σ)→TΣ→ℙTΣ. By composing with this map, any framing gives rise to a line field, all of whose winding numbers are even. Conversely, any line field with even winding numbers arises from a framing (see, for instance, [30, Lem. 1.1.4]).
The set of homotopy classes of line fields is an H1(Σ,ℤ)-torsor (see [30]).
Finally, we describe how a choice of spanning ribbon graph induces a line field on the complement of its vertices.
Example 4.8.
Let S be a marked surface with a spanning graph G. Then S\G0 inherits a canonical (homotopy class of a) line field νG, which we can depict locally near a vertex (of valency 5 for concreteness) of G as follows.
The winding number with respect to νG of an embedded loop wrapping clockwise around a vertex of valency m is thus given by −m.
Suppose that G→G′ is a contraction between spanning graphs of S, contracting a set E of edges of G. Then the corresponding line fields are closely related: the winding numbers of immersed loops not intersecting the edges in E are not affected by the contraction.
Perverse schobers
In this section, we recall the definition of perverse schober parametrized by a ribbon graph. For more background on perverse schobers on surfaces, see [10, § 3], which refines the treatment in [5, §§ 3 and 4].
Given n∈ℕ≥1, we denote by Gn the ribbon graph with a single vertex v and n incident external edges. We call Gn the n-spider.
Definition 4.9.
Let n≥1. An R-linear perverse schober parametrized by the n-spider, or on the n-spider for short, consists of the following data:
If n=1, an R-linear spherical adjunction
F:𝒱⟷𝒩:G,
that is, an adjunction whose twist functor
There is a different much-used convention for the definition of the twist and cotwist functors in the literature (see, for instance, [1, 39]). This convention differs from the one considered in this paper in two ways: firstly, instead of forming the cone of the unit, the other convention considers the cocone of the unit and similarly the cone of the counit. Furthermore, the labels of the twist and cotwist are swapped.
Let us illustrate our convention in the case of a spherical object: thinking of 𝒱 as the categorified vanishing cycles, this category is then given by 𝒱=𝒟(k). The twist functor acts on 𝒟(k) as a shift functor. The cotwist is the arising autoequivalence T𝒩 of the category 𝒩 containing the spherical object.
T𝒱=cof(id𝒱→unitGF)∈Fun(𝒱,𝒱) and cotwist functor T𝒩=fib(FG→counitid𝒩)∈Fun(𝒩,𝒩) are equivalences. Such functors F and G are also called spherical functors (see [1]). We also point out that such spherical functors F,G automatically admit all left and right adjoints, given by compositions of F and G with powers of the (co)twist functors (see [16]).
If n≥2, a collection of R-linear adjunctions
(Fi:𝒱n⟷𝒩i:Gi)i∈ℤ/nℤ
satisfying that
Gi is fully faithful, that is, FiGi≃id𝒩i via the counit,
Fi∘Gi+1 is an equivalence of ∞-categories,
Fi∘Gj≃0 if j≠i,i+1,
Gi admits a right adjoint radj(Gi), and
fib(radj(Gi+1))=fib(Fi) as full subcategories of 𝒱n.
We will also consider a collection of functors (Fi:𝒱n→𝒩i)i∈ℤ/nℤ as determining a perverse schober on the n-spider, or as a perverse schober on the n-spider for short, if there exist adjunctions (Fi⊣radj(Fi))i∈ℤ/nℤ which define a perverse schober on the n-spider. Such a collection of functors (Fi)i∈ℤ/nℤ can be equivalently encoded as a functor Fun(Exit(Gn),LinCatR).
One can show that the datum of a perverse schober on the n-spider is for any n≥1 equivalent to the datum of a perverse schober on the 1-spider. This gives rise to an explicit model of perverse schobers on the n-spider, which we describe in the following. This construction is related to Dyckerhoff’s categorified Dold–Kan correspondence [13] and the Waldhausen S•-construction (see also the discussion in [5]). Let F:𝒱→𝒩 be an R-linear spherical functor. We denote by 𝒱Fn the pullback in LinCatR of the following diagram.
Explicitly, the ∞-category 𝒱Fn consists of diagram
a→b1→⋯→bn−1,
where a∈𝒱, bi∈𝒩 and the morphism a→b1 lies in the Grothendieck construction of F. We define
ϱ1:𝒱Fn→𝒩
as the projection functor to bn−1. For 1<i<n, we define recursively ϱi as the doubly left adjoint of ϱi−1, which is the functor
ϱi=cofn−i+1,n−i+2[i−2]:𝒱Fn→𝒩
given by a suspension of the composite of the projection functor to bn−i→bn−i+1 with the cofiber functor. We define ϱn as the doubly left adjoint of ϱn−1, which maps an object (4.1) to the cofiber cof(F(a)→𝛼b1)[m−2]. We further define for 1≤i≤n the functor ςi as the right adjoint of ϱi. We thus have a sequence of adjunctions
ϱn⊣ςn⊣ϱn−1⊣⋯⊣ς2⊣ϱ1⊣ς1.
There is a further adjunction
ς1T𝒩−1[1−n]⊣ϱn,
where T𝒩 denotes the cotwist functor of the adjunction F⊣G (see [5, Lem. 3.8]).
Proposition 4.10
([5, 10]). Let F:𝒱→𝒩 be a spherical functor. The collection of adjunctions
(ϱi:𝒱Fn⟷𝒩:ςi)i∈ℤ/nℤ
define an R-linear perverse schober on the n-spider, denoted ℱn(F). Furthermore, for every R-linear perverse schober on the n-spider ℱ, there exist such a spherical functor F and an equivalence
ℱ≃ℱn(F)∈Fun(Exit(Gn),LinCatR).
We call any choice of such F the spherical functor underlying ℱ.
Let v be a vertex of valency n of a ribbon graph G. Let Exit(G)v/ be the undercategory, which has n+1 objects, which can be identified with v and its n incident halfedges and non-identity morphisms going from v to these halfedges. There is a functor Exit(G)v/→Exit(G), which is fully faithful if G has no loops incident to v.
Definition 4.11.
Let G be a ribbon graph. A functor ℱ:Exit(G)→LinCatR is called an R-linear G-parametrized perverse schober if for each vertex v of G, the restriction of ℱ to Exit(G)v/ determines a perverse schober parametrized by the n-spider in the sense of Definition 4.9.
We also call a functor Exit(G)→LinCatRdual or Exit(G)→LinCatRcpt−gen a G-parametrized perverse schober if its composite with LinCatRdual,LinCatRcpt−gen→LinCatR is a G-parametrized perverse schober.
Remark 4.12.
A G-parametrized perverse schober ℱ assigns to each edge of G an equivalent stable ∞-category, referred to as the generic stalk of ℱ, and usually denoted by 𝒩.
Definition 4.13.
Let ℱ be an R-linear G-parametrized perverse schober. For a vertex v of G, consider a choice of spherical functor Fv:𝒱v→𝒩 underlying the restriction of ℱ to Exit(G)v/ in the sense of Proposition 4.10. We call v a singularity of ℱ if 𝒱v≄0.
Definition 4.14.
Let ℱ be an R-linear G-parametrized perverse schober.
We denote by Γ(G,ℱ)=limLinCatR(ℱ) the limit of ℱ in the ∞-category LinCatR of R-linear ∞-categories. We call Γ(G,ℱ) the ∞-category of global sections of ℱ.
Suppose that ℱ takes values in compactly generated R-linear ∞-categories. We denote by Γdual(G,ℱ)=limLinCatRcpt−gen(ℱ) the limit of ℱ in the ∞-category LinCatRcpt−gen. We call Γdual(G,ℱ) the ∞-category of locally compact global sections.
Remark 4.15.
Recall that Γ(G,ℱ) agrees with the limit of ℱ in Cat∞ and can thus be identified with the ∞-category of coCartesian sections of the Grothendieck construction of ℱ, that is, the coCartesian fibration classified by ℱ (see [36, Prop. 05RX]).
Similarly, the ∞-category of locally compact global sections Γdual(G,ℱ) can be identified with the Ind-completion of the limit in Cat∞ of the pointwise restriction of ℱ to the subcategories of compact objects. Hence a compact object Y∈Γdual(G,ℱ)c consists of a pointwise compact coCartesian section of the Grothendieck construction.
Definition 4.16.
Given an R-linear G-parametrized perverse schober ℱ and an edge e of G, we denote by eve:Γ(G,ℱ)→ℱ(e) the evaluation functor, which maps a coCartesian section of the Grothendieck construction to its value at e.
The functor eve is R-linear and preserves limits and colimits, which can be proven using the fact that limits and colimits in the functor ∞-category of sections of the Grothendieck construction are computed pointwise.
We will often consider the product of the evaluation functors at the external edges
∏e∈G1∂eve:Γ(G,ℱ)→∏e∈G1∂ℱ(e).
By the ∞-categorical adjoint functor theorem, ∏e∈G1∂ev admits a right adjoint, which we denote by
∂ℱ:∏e∈G1∂ℱ(e)→Γ(G,ℱ).
The adjunction ∏e∈G1∂eve⊣∂ℱ is spherical (see [9, Thm. 5.2.5] for a (sketch of) proof).
Next, we briefly discuss how to relate perverse schober parametrized by different ribbon graphs. Given a ribbon graph G and an edge e of G which is not a loop, we can contract e to create a new ribbon graph G′. The two vertices incident to e are identified in G′, the edge e removed, but otherwise G′ is the same as G. More generally, we say that a ribbon graph G′ is a contraction of G if G′ can be obtained by contracting (automatically finitely many) edges of G. In this case, we write c:G→G′.
Proposition 4.17
([6, Prop. 4.28]). Let ℱ be an R-linear G-parametrized perverse schober and c:G→G′ a contraction which collapses no edge of G which is incident to two singularities of ℱ. Then there exists a canonical G′-parametrized perverse schober c∗(ℱ) together with an R-linear equivalence of ∞-categories
Γ(G,ℱ)≃Γ(G′,c∗(ℱ)).
We note that if ℱ above takes values in LinCatRcpt−gen, then
Γdual(G,ℱ)≃Γdual(G′,c∗(ℱ))
holds as well.
We end this section by describing the relationship between global sections and locally compact global sections of perverse schobers in typical situations.
Lemma 4.18.
Let G be a ribbon graph and ℱ:Exit(G)→LinCatRcpt−gen be an R-linear G-parametrized perverse schober. Suppose that for every vertex v of G, the spherical functor Fv:𝒱v→𝒩 underlying ℱ near v is conservative.
A global sectionY∈Γ(G,ℱ)is finite in the sense of Definition2.24if and only ifeve(Y)∈ℱ(e)is finite for all edgese∈G1.
Suppose the generic stalk𝒩is proper and thatFvreflects compact objects for all verticesvof G. Then the locally compact global sections ofℱcoincide with theInd-completion of the finite global sections, that is,
Γdual(G,ℱ)≃IndΓ(G,ℱ)fin.
Proof.
We begin by proving part (1). For an edge e∈G1, denote the left adjoint of the evaluation functor eve:Γ(G,ℱ)→ℱ(e)≃𝒩 by eve∗. A straightforward computation shows that the assumption that Fv:𝒱v→𝒩 is conservative for all vertices v with incident edges e1,…,em is equivalent to the assertion that
∏i=1mℱ(v→ei):ℱ(v)⟶∏i=1mℱ(ei)
is conservative. Thus, a global section Y of ℱ vanishes if and only if eve(Y)≃0 for all e∈G1.
Let {Xi}i∈I be a collection of compact generators of the generic stalk 𝒩 of ℱ, meaning that an object N∈𝒩 vanishes if and only if Mor𝒩(Xi,N)≃0 for all i∈I. The collection of objects {eve∗(Xi)}i∈I,e∈G1 compactly generates Γ(G,ℱ), as follows from the equivalence
MorΓ(G,ℱ)(eve∗(Xi),Y)≃Mor𝒩(Xi,eve(Y)).
Hence, the objects eve∗(Xi) suffice to test the finiteness of global sections. A global section Y∈Γ(G,ℱ) is thus finite if and only if eve(Y)∈ℱ(e)≃𝒩 is finite for all e∈G1.
We proceed with proving part (2). Let Y∈Γ(G,ℱ) with eve(Y)∈ℱ(e)c for all e∈G1. The assertion that Fv reflects compact objects for a vertex v, with incident edges e1,…,em, is equivalent to the assertion that the functor ∏i=1mℱ(v→ei) reflects compact objects. It follows that Y is a pointwise compact coCartesian section of the Grothendieck construction, that is, it lies in Γdual(G,ℱ)c.
Since 𝒩 is proper, we have 𝒩fin≃𝒩c. Combining part (1) with the above thus shows that the finite global sections are the locally compact global sections, that is, Γ(G,ℱ)fin≃Γdual(G,ℱ)c. Passing to Ind-completions shows part (2).∎
Remark 4.19.
Let G be a spanning graph of a marked surface S, and let ℱ:Exit(G)→LinCatRcpt−gen be a G-parametrized perverse schober with non-vanishing generic stalk. One can show that Γdual(G,ℱ)≃Γ(G,ℱ) if and only if each boundary component of the marked surface contains at least one marked point.
Monodromy of perverse schobers
Let G be a spanning graph of a marked surface S and ℱ an R-linear G-parametrized perverse schober. We denote by P the set of vertices of G which are singularities of ℱ (see Definition 4.13). In analogy with perverse sheaves on S, which restrict to a local system of abelian groups or vector spaces on S\P, one may wish to associate a monodromy local system of R-linear ∞-categories to ℱ. It turns out that there is indeed a reasonable notion of monodromy of a perverse schober, but it does not canonically assemble into a local system on S\P. Instead, to define the local system S\P, one needs to input a choice of framing of S\P. Applying K0, we obtain the usual local system of abelian groups of the underlying perverse sheaf; the choice of framing does not affect this local system of abelian groups.
To define the monodromy local system, we will in the first step define an auxiliary local system of transports on S\G0. From this, we will obtain a local system on the projectivized tangent bundle ℙT(S\G0) with monodromy [1] on the circle fiber. We then pull back the local system to the frame bundle FrS\G0. Any choice of framing then allows us to further pull back this local system to a local system on S\G0, and this local system extends to S\P, as desired.
To obtain the transport along a loop, we compose local transports. We define these in Construction 4.21. For technical convenience, we will replace in the following the surface S with spanning graph G by the homotopic non-compact surface ΣG described in Remark 4.20.
Remark 4.20.
Let G be a ribbon graph. To each vertex v of G of valency n, we associate a (non-compact) surface, denoted Σv, or also Σn, with an embedding of v and its n incident halfedges. We depict Σv as follows (in green). The dotted lines correspond to open ends, whereas the solid lines indicate the boundary.
We define the thickening of G to be the surface ΣG, obtained by gluing the surfaces Σv, whenever two vertices are incident to the same edge at their boundary components corresponding to the edge. The surface ΣG comes with an embedding of |G|, which is also a homotopy equivalence. We define the subset B⊂ΣG as the union of the images of the boundaries ∂Σv for all vertices v. Note that each edge e∈G1 of G intersects exactly one connected component of B exactly once; we denote this component by Be⊂B.
Construction 4.21.
Let n≥2; the case n=1 is addressed at the end. The n-spider Gn is embedded in Σn=Σv; we denote its central vertex by v. Consider an embedded curve δ:[0,1]→Σv\{v} satisfying that δ(0),δ(1)∈∂Σv and that the boundary component Be1 of Gn containing δ(1) lies one step in the counterclockwise direction of the boundary component Be0 containing δ(0). We can depict this setup as follows.
Given an R-linear Gn-parametrized perverse schober ℱ, we define the transport ℱ→(δ) of ℱ along δ as the R-linear equivalence
ℱ(e0)→ladj(ℱ(v→e0))ℱ(v)→ℱ(v→e1)ℱ(e1).
Reversing the orientation of δ yields a path δrev going one step in the clockwise direction around v, and the transport of ℱ along δrev is defined as
ℱ→(δrev)≔ℱ(v→e0)∘radj(ℱ(v→e1)):ℱ(e1)⟶ℱ(e0).
We thus have ℱ→(δrev)≃(ℱ→(δ))−1.
Consider now an arbitrary curve δ:[0,1]→Σv\{v} satisfying that δ(0),δ(1)∈∂Σv. Then δ is homotopic relative to its endpoints either to a curve contained in the boundary ∂Σv, in which case we set i=0, or there exists i∈ℤ\{0} such that δ is homotopic relative to its endpoints to the composite δ|i|∗⋯∗δ1 of |i| embedded paths δ1,…,δ|i| as above, each wrapping one step counterclockwise if i>0 0\)]]> and one step clockwise if i<0. Thus, δ goes i∈ℤ steps counterclockwise. We define the transport along δ as
ℱ→(δ)≔{idℱ(e0),i=0,ℱ→(δ|i|)∘⋯∘ℱ→(δ1),i≠0.
If i=n, that is, δ is a loop wrapping once counterclockwise around v, it follows from adjunctions (4.2) and (4.3) that ℱ→(δ)≃T𝒩−1[1−n], with T𝒩−1 the inverse cotwist of the spherical adjunction underlying ℱ at v.
We conclude with the case n=1. Consider a perverse schober ℱ parametrized by the 1-spider, with vertex v and edge e, that is, a spherical functor F:𝒱=ℱ(v)→𝒩=ℱ(e). Let δ:[0,1]→Σ1\{v} be a curve with δ(0),δ(1)∈∂Σ1. If δ wraps i∈ℤ times counterclockwise around v, we define the transport as
where T𝒩 denotes the cotwist functor of the adjunction F⊣radj(F).
Example 4.22.
Consider an R-linear perverse schober ℱ on the m-spider Gm for some m≥2, with central vertex v, generic stalk 𝒩 and no singularity at v. Such a perverse schober categorifies a perverse sheaf without singularities (i.e., a local system) on the disc which automatically has trivial monodromy.
The perverse schober ℱ is up to equivalence described by the adjunctions
(ϱi[−i]:Fun(Δm−1,𝒩)⟷𝒩:ςi[i])1≤i≤m.
The transport functors along paths wrapping one step clockwise around v from the (i+1)-th edge of Gm to the i-th edge are given by
ϱi[−i]∘ςi+1[i+1]≃[1],
for 1≤i≤m−1, as follows from adjunction (4.2). For i=m, the transport is given by
ϱm[−m]∘ς1[1]≃[−1],
as follows from adjunction (4.3) using that the cotwist functor of the adjunction 0↔𝒩 is given by T𝒩≃[−1]. The transport of an embedded full loop enclosing v going in the clockwise direction is thus given by [m−2].
Construction 4.23.
Let G be a ribbon graph and consider a curve η:[0,1]→ΣG\G0 going from Be0 to Be1 for some e0,e1∈G1. We can write η uniquely as the composite of a minimal number of curves δ1,…,δm with endpoints in B such that each δi is contained in Σvi⊂ΣG for some vertex vi∈G0. Concretely, the paths δi near a given vertex v are obtained as the components of the intersection of γ with Σv⊂ΣG.
Let ℱ be an R-linear G-parametrized perverse schober. We define the transport ℱ→(η) of ℱ along η as the R-linear equivalence
ℱ→(δm)∘⋯∘ℱ→(δ1):ℱ(e0)⟶ℱ(e1).
The following lemma collects some properties of the transport functors.
Lemma 4.24.
Let ℱ be an R-linear G-parametrized perverse schober.
Consider two curvesη,η′:[0,1]→ΣG\G0withη1(0)=η2(0)∈Be0,η1(1)=η2(1)∈Be1fore0,e1∈G1. Ifη,η′are homotopic inΣG\G0with the homotopy fixing endpoints, then
ℱ→(η1)≃ℱ→(η2).
Letη,η′:[0,1]→ΣG\G0be curves with endpoints in and such that B η(1)=η′(0). Denote their composite byη′∗η. Then
ℱ→(η′∗η)≃ℱ→(η′)∘ℱ→(η).
Letc:G→G′be a contraction of ribbon graphs contracting a setE⊂G1of edges, none connecting two singularities ofℱ. Letη:[0,1]→ΣG\G0be a curve with endpoints inB\⋃e∈EBenot intersecting any edges inE. Choose a smooth mapC:ΣG→ΣG′that realizes the contraction|G|→|G′|by contracting small neighborhoods of the edges inEand that restricts on the complement of these neighborhoods to a diffeomorphism. ThenC∘ηis a curve inΣG′\G0′and
ℱ→(η)≃c∗(ℱ)→(C∘η).
Proof.
Parts (1) and (2) are immediate from the definition of transport. For part (3), it suffices to consider the case that c contracts a single edge e, since any contraction is a finite composition of such contractions. As ℱ and c∗(ℱ) are identical away from a neighborhood of e, it suffices to show that the transports near e of ℱ and c∗(ℱ) coincide. Using the local model for ℱ from Proposition 4.10 and the local model for c∗(ℱ) described in [6, Lem. 4.26], this is straightforward to verify.∎
For the following, we fix a ribbon graph G, and we choose once and for all an edge e∈G1. Let x∈e∩Be be the intersection point.
Lemma 4.25.
Let ℱ be an R-linear G-parametrized perverse schober. The assignment γ↦ℱ→(γ) on based loops at x defines a group homomorphism
ℱ→:π1(ΣG\G0,x)⟶π0Aut(ℱ(e)),
that is, a local system on ΣG\G0, with values in the group of equivalence classes of R-linear autoequivalences of ℱ(e).
Proof.
This assignment is well defined by part (1) of Lemma 4.24 and a group homomorphism by part (2) of Lemma 4.24.∎
The main issue with the local system from Lemma 4.25 is that it does not extend to ΣG\P, where P denotes the set of singularities of ℱ (see Example 4.22). This is in general not even true on K0.
Suppose x′∈ℙT(ΣG\G0) is an inverse image of x under the bundle projection ℙT(ΣG\G0)→ΣG\G0. Consider the short exact sequence of groups
1→π1(ℙTxΣG\G0,x′)→π1(ℙTΣG\G0,x′)→π1(ΣG\G0,x)→1,
where π1(ℙTxΣG\G0)≃ℤ (with 1 corresponding to the counterclockwise half-turn). The line field νG from Example 4.8 defines a splitting
We define the local system ℒℱ on Fr(ΣG\G0) as the pullback of ℒℙTℱ along the map Fr(ΣG\G0)→ℙT(ΣG\G0).
Remark 4.27.
Let v be a vertex of G of valency m. Near v, the local system ℒℱ can be described as follows. Since Σv is diffeomorphic to a subset of ℝ2, there exist canonical splittings ℙTΣv≃Σv×S1 and Fr(Σv)≃Σv×GL(2,ℝ), which restrict to splittings
ℙT(Σv\{v})≃(Σv\{v})×S1
and
Fr(Σv\{v})≃Σv\{v}×GL(2,ℝ).
Let γ:S1→Σv\{v} be an embedded loop mapping the base point to ∂Σv and wrapping one time clockwise around v. Then γ defines loops γ′ in (Σv\{v})×S1 and γ′′ in Σv\{v}×GL(2,ℝ), each of which is constant in the second component. We have
ℒℱ(γ′′)≃ℒℙTℱ(γ′)≃ℱ→(γ)[W(γ)]=ℱ→(γ)[−m].
By Example 4.22, if v is not a singularity of ℱ, then
ℒℱ(γ′′)≃[−2].
Proposition 4.28.
Let ℱ be an R-linear G-parametrized perverse schober with singularities at P, and let ξ be a framing on ΣG\P. The pullback local system ξ∗ℒℱ then extends to a local system
ξ∗ℒℱ:π1(ΣG\P,x)⟶π0Aut(ℱ(e)).
We call this local system the monodromy of ℱ with respect to the framing ξ.
Proof.
Consider the restriction ξ|ΣG\G0. Then the winding number of a clockwise embedded loop wrapping one time around a non-singular vertex v∈G0\P is given by −2. Comparing with (4.7), we see that the monodromy of ξ|ΣG\G0∗ℒℱ along this loop is trivial. It follows that ξ|ΣG\G0∗ℒℱ extends to the desired local system on ΣG\P.∎
Remark 4.29.
The above constructions can be seen as implementing the following observation: the group H1(ΣG\G0,ℤ) acts on the ∞-category of local systems on ΣG\G0. The line field νG from Example 4.8 gives a base point of the H1(ΣG\G0,ℤ)-torsor of homotopy classes of line fields. Thus, the line field arising from a framing ξ acts on the local system ℱ→ to produce a new local system, given by ξ∗ℒℱ. Contrary to ℱ→, the local system ξ∗ℒℱ extends to ΣG\P.
Example 4.30.
Consider a perverse schober ℱ on the disc D with one boundary marked point, parametrized by the 1-spider G1 with vertex v. Thus, ℱ consists of a spherical functor F:𝒱→𝒩. Let G be the right adjoint of F and T𝒩 the cotwist functor of F⊣G. Choosing any framing on D and restricting it to D\{v}, the corresponding clockwise monodromy of ℱ around v is given by T𝒩[1]. Passing to Grothendieck groups, that is, applying K0(-), we obtain a perverse sheaf on D with (at most) one single singularity. We have
K0(T𝒩)=K0(F)K0(G)−K0(id𝒩).
The automorphism of K0(𝒩)
K0(T𝒩[1])=−K0(T𝒩)=K0(id𝒩)−K0(F)K0(G)
describes the usual monodromy of this perverse sheaf.
Remark 4.31.
Let ℱ be an R-linear G-parametrized perverse schober and suppose that ℱ(e) is 2n-periodic for some n≥1, in the sense that [2n]≃idℱ(e). Let x′′∈Fr(ΣG\G0) be an inverse image of x∈ΣG\G0. The fiberwise monodromy of ℒℱ, that is, the action of 1∈ℤ≃π1(Frx(ΣG\G0),x′′) is given by [2].
Suppose first that n=1, that is, ℱ(e) is 2-periodic. Then the local system
ℒℱ:π1(Fr(ΣG\G0),x′′)⟶π0Aut(ℱ(e))
factors through the quotient
π1(Fr(ΣG\G0),x′′)/π1(Frx(ΣG\G0),x′′)≃π1(ΣG\G0,x)
and thus canonically defines a local system on ΣG\G0 (that even extends to ΣG\P), without a choice of framing.
Now suppose that n>1 1\)]]>. An n-spin structure on ΣG\G0 amounts to an n-fold connected covering of Fr(ΣG\G0). Pulling back ℒℱ along this covering yields a local system with trivial monodromy on the fiber over ΣG\G0 that hence restricts as in the case n=1 to a local system on ΣG\P.
Remark 4.32.
Let ℱ be an R-linear G-parametrized perverse schober with singularities at P⊂G0. Applying any additive invariant E, such as Hochschild homology, to the local system ℒℱ on Fr(ΣG\G0) yields a local system with trivial monodromy on the fiber, since E([2])≃id. By the same argument as in Remark 4.31, E(ℒℱ) thus defines a local system on ΣG\P.
The monodromy of a perverse schober is independent of the choice of ribbon graph in the appropriate sense.
Lemma 4.33.
Let ℱ be an R-linear G-parametrized perverse schober with singularities at P. Let c:G→G′ be a contraction of ribbon graphs not contracting the edge e and no edges connecting any two vertices in P and choose a smooth map C:ΣG→ΣG′ as in Lemma4.24. Let ξ′ be a framing on ΣG′\P, and let ξ=C∗(ξ′) be the corresponding framing on ΣG\P. There exists an equivalence between ξ∗ℒℱ and the local system
This follows from Lemma 4.24 and the relation between the line fields νG and νG′ described in Example 4.8.∎
Next, we prove that a perverse schober without singularities is completely determined by its generic stalk and monodromy.
Proposition 4.34.
Let ℱ1,ℱ2 be two R-linear G-parametrized perverse schobers without singularities. Let ξ be a framing of ΣG. Assume that ℱ1(e)=ℱ2(e). The following two are equivalent:
There exists an equivalence of G-parametrized perverse schobersℱ1≃ℱ2.
The monodromy local systemsξ∗ℒℱ1andξ∗ℒℱ2are equivalent.
Proof.
It is clear that (1) implies (2). We next show that (2) implies (1). Denote by 𝒩=ℱ1(e) the generic stalk. Choose a contraction c:G→G′ such that e is not contracted and G′ has only a single vertex. Part (3) of Lemma 4.24 implies that the local systems of transport of c∗(ℱ1) and c∗(ℱ2) are equivalent.
We choose a total order of the m halfedges incident to the vertex v of G′, compatible with their given (counterclockwise) cyclic order. We denote the i-th halfedge by ai and its corresponding edge by ei. We can replace c∗(ℱ1) and c∗(ℱ2) by equivalent G′-parametrized perverse schobers, denoted 𝒢1 and 𝒢2, such that for j=1,2 and 1≤i≤m,
𝒢j(v)=𝒱0𝒩m=Fun(Δm−2,𝒩),𝒢j(ei)=𝒩
and
𝒢j(v→aiei)=Si,j∘ϱi[−i],
where Si,j is some autoequivalence of 𝒩. The monodromy relative ξ along a path γ starting at ei and going around a given loop of G′, composed of halfedges ai,ai′ of 𝒢j, is given by Si′,j∘Si,j−1[i−i′+W(γ)] if i′>i i\)]]> and by Si′,j∘Si,j−1[i−i′−m+2+W(γ)] if i′<i. We thus find Si′,1Si,1−1≃Si′,2Si,2−1. We can additionally assume that Si′,2=Si′,1=id𝒢(e) by replacing 𝒢1,𝒢2 by equivalent perverse schobers once more. We thus conclude Si,1≃Si,2 as well. Performing this argument for all loops of G′ shows that 𝒢1≃𝒢2, concluding the proof.∎
The next example illustrates that a perverse schober consists of more data than its singularity data and monodromy data.
Example 4.35.
Consider the following spanning graph G of the 1-gon.
Consider a G-parametrized perverse schober ℱ with two singularities at the vertices labeled × and no singularity at the vertices labeled ∘. Let F1:𝒱1→𝒩 and F2:𝒱2→𝒩 be the underlying spherical adjunctions at the two vertices. These are determined only up to pre- and postcomposition with equivalences of ∞-categories. The perverse schober ℱ thus corresponds, up to equivalence, to a diagram
with S1,S2 autoequivalences of 𝒩. Note that Fun(Δ1,𝒩)≃𝒱0𝒩3, with 0𝒩:0→𝒩 the spherical zero functor. Composing with the autoequivalence Fun(Δ1,S2−1) of Fun(Δ1,𝒩) replaces S1 by S2−1S1 and S2 by id𝒩. Up to equivalence of perverse schobers, we may thus assume that S2=id𝒩. In total, the equivalence class of the perverse schober ℱ is determined by the functors S1,F1 and F2, up to natural equivalence of these functors and precomposition of F1,F2 with autoequivalences of 𝒱1,𝒱2.
We equip the 1-gon with any framing (it is unique up to homotopy) and denote its restriction to the complement of the two singular vertices by ξ. The monodromy of ℱ with respect to ξ along a clockwise loop wrapping once around the left singularity is given by S1T𝒩S1−1[1], with T𝒩 the cotwist of F1⊣radj(F1), which is the suspension of the cotwist functor of the adjunction S1F1⊣radj(F1)S1−1. The monodromy of the clockwise loop wrapping once around the right singularity is similarly given by the cotwist functor of the adjunction F2⊣radj(F2).
Since the choice of S1 affects the monodromy of ℱ only by conjugation with S1, we find that, given the functors F1,F2 and the monodromy local system, one cannot in general recover the autoequivalence S1, and thus also cannot recover the equivalence class of the perverse schober. One can make this failure explicit by choosing, for instance, F1=F2=ϕ∗ the functor from Section 6.3 for n=3, where T𝒩=[3], and S1=φ∗ is the functor from the proof of Theorem 6.9.
Calabi–Yau structures and perverse schobers
We begin with the local picture in Section 5.1 by constructing relative Calabi–Yau structures on the sections of perverse schobers parametrized by the n-spider. In Section 5.2, we discuss the construction of relative Calabi–Yau structures on the global sections of perverse schobers. Finally, in Section 5.3, we return to the case of perverse schobers parametrized by the 1-spider, that is, spherical adjunctions, and give a novel and easy-to-verify criterion for the existence of a weak right Calabi–Yau structure.
Calabi–Yau structures locally
Consider a perverse schober on the m-spider ℱ. Proposition 4.10 states that ℱ arises from a spherical adjunction F:𝒟↔𝒞:G, in the sense that ℱ is given, up to equivalence, by the collection of adjunctions
(ϱi:𝒱Fm⟷𝒞:ςi)1≤i≤m.
We can combine these adjunctions into a single adjunction
R=(ϱ1,…,ϱm):𝒱Fm⟷𝒞×m:S=(ς1,…,ςm).
As it turns out, this adjunction is spherical, and its twist functor describes the change in the perverse schober on the m-gon when rotating it by an angle of 2π/m (see Proposition 5.1). Note that this is not a genuine symmetry; a full rotation does not return the perverse schober to itself, but instead changes it by the monodromy around the singularity, that is, the suspension of the cotwist functor of F⊣G.
Proposition 5.1
([5, Props. 3.8 and 3.11]). The adjunction R⊣S is spherical. Let T𝒱Fm be its twist functor and T𝒞 the cotwist functor of the adjunction F⊣radj(F). There exist equivalences of functors
ϱi∘T𝒱Fm≃{ϱi+1for 1≤i≤m−1,T𝒞[m−1]∘ϱ1for i=m.
The goal of this section is to prove the following proposition, roughly stating that the functors R,S can be equipped with Calabi–Yau structures, provided that F,G are already equipped with relative Calabi–Yau structures. In this case, the Serre or inverse dualizing functors id𝒱Fm∗,id𝒱Fm! thus describe the effect of partial rotation of the m-gon.
To simplify the signs, we change R and S in the following proposition by composition with componentwise suspension functors.
Proposition 5.2.
Let F:𝒟↔𝒞:G be a spherical adjunction of dualizable R-linear ∞-categories and m≥1. Consider the spherical adjunction
Suppose that𝒞,𝒟are smooth and that there exists a classηG:R[n]→HH(𝒟,𝒞)S1defining a leftn-Calabi–Yau structure onG, which restricts to a left(n−1)-Calabi–Yau structureη𝒟:R[n−1]→HH(𝒞)S1on𝒞. Then the functorSFmadmits a canonical leftn-Calabi–Yau structure, which restricts on𝒞×mto
η𝒞×m:R[n−1]⟶HH(𝒞×m)S1≃⨁mHH(𝒞)S1.
Suppose that𝒞,𝒟are proper and assume that there exists a classηF:R[n]→HH(𝒟,𝒞)S1∗defining a rightn-Calabi–Yau structure onF, which restricts to a right(n−1)-Calabi–Yau structureη𝒞:R[n−1]→HH(𝒞)S1∗on𝒞. Then the functorRFmadmits a canonical rightn-Calabi–Yau structure, which restricts on𝒞×mto
η𝒞×m:R[n−1]⟶HH(𝒞×m)S1∗≃⨁mHH(𝒞)S1∗.
Using the gluing properties of Calabi–Yau structures, we will reduce the proof of Proposition 5.2 to the case m=3 and 𝒟=0. This case is then first solved for 𝒞=RModR, which admits canonical left and right 0-Calabi–Yau structures and then for arbitrary 𝒞 by tensoring (see Proposition 3.9).
Construction 5.3.
Let 𝒞=RModR, considered as an R-linear smooth and proper ∞-category. We construct two inverse equivalences U,U−1:Fun(Δ1,𝒞)→Fun(Δ1,𝒞) via Kan extension.
Consider the ∞-category 𝒳 of diagrams in 𝒞 of the following form, where all squares are biCartesian, that is, both pushout and pullback squares.
One can formally characterize the ∞-category 𝒳 as consisting of diagrams which are repeated Kan extensions of their restriction to a→b. The restriction functor to a→b thus defines by [31, Prop. 4.3.2.15] a trivial fibration ϕ:𝒳→Fun(Δ1,𝒞). It hence admits an inverse, unique up to contractible space of choices. We denote by ψ′:𝒳→Fun(Δ1,𝒞) the restriction functor to a′→b′ and by ψ′′:𝒳→Fun(Δ1,𝒞) the restriction functor to a′′→b′′.
The functor U is defined as the suspension of the composite functor
Fun(Δ1,𝒞)→ϕ−1𝒳→ψ′Fun(Δ1,𝒞),
and the functor U−1 is defined as the looping of the composite functor
Fun(Δ1,𝒞)→ϕ−1𝒳→ψ′′Fun(Δ1,𝒞).
Remark 5.4.
An alternative description of the functor U from Construction 5.3 is as follows. Let again 𝒞=RModR. We have functors ev1,cof:Fun(Δ1,𝒞)→𝒞, together with a natural transformation ev1→cof, as well as the further fully faithful functors ι0,ι1:𝒞→Fun(Δ1,𝒞), given informally by ι0:c↦(c→0) and ι2:c↦(0→c), together with a natural transformation ι1[−1]→ι2. Composing these functors, we have an induced natural transformation ι1ev1[−1]→ι2cof, whose cofiber describes an endofunctor of Fun(Δ1,𝒞), given by the assignment
(a⟶b)↦(b⟶cof(a→b)).
This functor is equivalent to the endofunctor U, as follows from the universal property of the R-linear ∞-category Fun(Δ1,𝒞), as the lax limit of the functor id𝒞:𝒞→𝒞, considered as a Δ1-indexed diagram in the (∞,2)-category of R-linear ∞-categories.
Lemma 5.5.
Let 𝒞=RModR. The R-linear ∞-category Fun(Δ1,𝒞) is smooth and proper, and the functor U from Construction5.3is a Serre functor of Fun(Δ1,𝒞).
Proof.
It is straightforward to check that Fun(Δ1,𝒞) is smooth and proper. Any object a→b in Fun(Δ1,𝒞) is given as the cofiber of a map (a[−1]→0)→(0→b). By the exactness of MorFun(Δ1,𝒞)(-,-) in the second entry, it follows that
which are functorial in a→b∈Fun(Δ1,𝒞)c,op, a′→b′∈Fun(Δ1,𝒞)c. The second-to-last equivalence arises in the same way as equivalence (5.2). The last equivalence uses the fact that the sequences a′→b′→cof(a′→b′) and b→cof(a→b)→a[1] are fiber and cofiber sequences, as well as Remark 5.4. This shows that U is a Serre functor, which concludes the proof.∎
Lemma 5.6.
Let 𝒞=RModR. We let η𝒞:R→HH(𝒞)S1 and η𝒞∗:R→HH(𝒟)S1∗ denote the lifts
Canonical lifts exist by Remark 2.27.
of the apparent classes in HH(𝒞)≃R and HH(𝒞)∗≃R. Note that η𝒞 and η𝒞∗ describe left and right 0-Calabi–Yau structures on 𝒞, respectively.
TheR-linear functor
S≔(ς1[1],ς2[2],ς3[3]):𝒞×3⟶Fun(Δ1,𝒞)
admits a unique left1-Calabi–Yau structure which restricts to the left0-Calabi–Yau structureη𝒞×3on𝒞×3.
TheR-linear functor
(ϱ1[−1],ϱ2[−2],ϱ3[−3]):Fun(Δ1,𝒞)⟶𝒞×3
admits a unique right1-Calabi–Yau structure which restricts to the right0-Calabi–Yau structure(η𝒞∗)×3on𝒞×3.
Proof.
We only prove part (1); part (2) can be proven similarly. The split localization sequence
𝒞→ς2[2]Fun(Δ1,𝒞)→ϱ3[−3]𝒞
furnishes us with a splitting HH(Fun(Δ1,𝒞))S1≃HH(𝒞)S1⊕HH(𝒞)S1. Applying the adjunctions ϱ3[−3]⊣ς3[3] and ς2[2]⊣ϱ1[−2], we observe that HH(ϱ3[−3])S1 and HH(ϱ1[−2])S1 are the two projection maps (HH(𝒞)S1)⊕2→HH(𝒞)S1 and that HH(ς3[3])S1,HH(ς2[2])S1 are the two inclusion maps HH(𝒞)S1→(HH(𝒞)S1)⊕2 of the direct sum decomposition.
where ℳ=(−110−101). The formula for ℳ follows from
ϱ1[−2]∘ς1[1]≃[−1]
and
ϱ3[−3]∘ς1[1]≃[−1].
We let η be the class R[1]→η𝒞HH(𝒞)S1[1]≃HH(Fun(Δ1,𝒞),𝒞×3)S1. The observation that ℳ(1,1,1)=0 implies that η indeed restricts to η𝒞×3 on 𝒞×3. Furthermore, η is clearly unique with this property.
Let σ𝒞=1∈R≃HH(𝒞) be the Hochschild class underlying η𝒞 and σ=(σ𝒞,σ𝒞,σ𝒞) the Hochschild class underlying η. To complete the proof, it remains to show that σ is non-degenerate. The class σ determines the diagram
together with a null-homotopy of the composite functor idFun(Δ1,𝒞)!→idFun(Δ1,𝒞). Composing the first two morphisms in the above diagram, we obtain the sequence
idFun(Δ1,𝒞)!⟶S!(id𝒞×3)→cuidFun(Δ1,𝒞).
The morphism u~ is by Lemma 2.34 equivalent to the composite of the unit of the adjunction SL⊣S with idFun(Δ1,𝒞)!. By Lemmas 2.21, 2.22 and 5.5, there exists an equivalence between idFun(Δ1,𝒞)! and the functor U−1 from Construction 5.3.
One can check that the functor T is furthermore equivalent to the cotwist functor of the adjunction S⊣SR. There thus exists a fiber and cofiber sequence
T→u′SSR→cuidFun(Δ1,𝒞),
where u′ is up to equivalence a unit of the adjunction SL⊣S composed with T (see [16] or [7, Rem. 2.10]). The respective counit maps in (5.3) and (5.4) describe the same counit map. The respective unit maps are also equivalent, up to composition with an autoequivalence of SSR. To show that this autoequivalence may be chosen trivially, we inspect the R-module of all possible autoequivalences of SSR=S!(id𝒞×3). We have
where ladj(S!)=SL(-)SRR, with SRR the right adjoint of SR. Since SR≃SL∘T, we have SRR≃T−1∘S and thus
ladj(S!)S!(id𝒟×3)≃SLSSRSRR≃SLSSLS.
The functor SLS splits as
SLS≃id𝒞×3⊕P,
where P is the twist functor of the adjunction SL⊣S. It acts via rotation, meaning that P sends the i-th component of the direct sum to the (i−1)-th component of the direct sum for all i∈ℤ/3ℤ and then acts with some suspensions on the three components. There are no non-zero natural transformations between id𝒞×3 and P or P2. It follows that the morphism
is an equivalence. We thus find that every possible autoequivalence of SSR can be accommodated by choosing a different Hochschild class of 𝒞×3. We may hence conclude from the existence of cofiber sequence (5.4) that there exists some choice of Hochschild class σ′∈R≃HH(Fun(Δ1,𝒞),𝒞×3)S1[−1] which restricts to the class (σ′)×3∈R⊕3≃HH(𝒞×3), which turns (5.3) into a cofiber sequence. Since ladj(S!)S! contains the identity as a direct summand, we find that S! is a conservative functor. The fact that σ′ induces an equivalence S!(id𝒞×3!)≃S!(id𝒞×3) thus implies that (σ′)×3:id𝒞×3!→id𝒞×3 is already an equivalence. It follows that σ′∈π0(R) must be an invertible element. Upon composing σ′ with its inverse in the ring π0(R), cofiber sequence (5.3) clearly remains a cofiber sequence. This shows that η already describes a left 1-Calabi–Yau structure, concluding the proof.∎
Proof of Proposition <xref rid="j_JNCG663_stat_005_127" ref-type="statement">5.2</xref>.
We only prove part (2); the proof of part (1) is analogous. We suppose that F:𝒟→𝒞 is a spherical functor with a right n-Calabi–Yau structure restricting to a right (n−1)-Calabi–Yau structure on 𝒞.
We first assume that F=0𝒞:0→𝒞. As shown in [5, Lem. 4.26], there is a pullback diagram in LinCatR
such that the functor ϱi[−i]:𝒱0𝒞m→𝒞 factors for i=1,2 as
𝒱0𝒞m→𝒱0𝒞3→ϱi[−i]𝒞
and for i=3,…,m as
𝒱0𝒞m→𝒱0𝒞m−1→ϱi−2[2−i]𝒟.
To show that R𝒞m admits the desired right n-Calabi–Yau structure, it thus suffices by Theorem 3.16 to show this in the case m=3. This case follows from combining Lemma 5.6 and Proposition 3.9.
Let now the above functor F:𝒟→𝒞 again be arbitrary. Again, by [5, Lem. 4.26], there exists a pullback diagram in LinCatR.
By Theorem 3.16, the above-constructed right n-Calabi–Yau structure on R0𝒟m+1 glues with the n-Calabi–Yau structure on F to the desired right n-Calabi–Yau structure on RFm.∎
Calabi–Yau structures on global sections
We begin by recording a direct consequence of the gluing property of Calabi–Yau structures.
Theorem 5.7.
Let ℱ:Exit(G)→LinCatRdual be a G-parametrized perverse schober.
Suppose thatℱtakes values in smoothR-linear∞-categories. Suppose further that
For each vertexvof G with incident halfedgesa1,…,amand corresponding edgese1,…,em, the functor
∏i=1mladj(ℱ(v→aiei)):∏i=1mℱ(ei)⟶ℱ(v)
carries a leftn-Calabi–Yau structure
ηv:R[n]→HH(ℱ(v),∏i=1mℱ(ei))S1.
We denote the restriction ofηvalong the functorladj(ℱ(v→aiei))by
ηe,ai:R[n−1]→HH(ℱ(ei))S1.
For each internal edgeeof G with incident halfedgesa≠b, we haveηe,a≃−ηe,b.
Then theR-linear∞-category of global sectionsΓ(G,ℱ)is smooth, and the functor from equation (4.4)
∂ℱ:∏e∈G1∂ℱ(e)⟶Γ(G,ℱ)
admits a leftn-Calabi–Yau structure.
Suppose thatℱtakes values in properR-linear∞-categories. Suppose further that
For each vertexvof G with incident halfedgesa1,…,amand corresponding edgese1,…,em, the functor
∏i=1mℱ(v→aiei):ℱ(v)⟶∏i=1mℱ(ei)
carries a rightn-Calabi–Yau structure
ηv:R[n]→HH(ℱ(v),∏i=1mℱ(ei))S1∗.
We denote the restriction ofηvalong the functorℱ(v→aiei)by
ηe,ai:R[n−1]→HH(ℱ(ei))S1∗.
For each internal edgeeof G with incident halfedgesa≠b, we haveηe,a≃−ηe,b.
Then the evaluation functor at the external edges
∏e∈G1∂eve:Γdual(G,ℱ)⟶∏e∈G1∂ℱ(e)
admits a rightn-Calabi–Yau structure.
Proof.
Part (ii) follows from repeated application of Theorem 3.16 by using that we can compute the limit over Exit(G) via repeated pullbacks. Part (i) follows from a similar argument from Theorem 3.15 by using that Γ(G,ℱ) is equivalent to the colimit in LinCatR of the left adjoint diagram of ℱ.∎
Given a parametrized perverse schober without singularities in the sense of Definition 4.13, also called a locally constant perverse schober, whose generic stalk admits a Calabi–Yau structure, Theorem 5.8 states that its global sections admit a Calabi–Yau structure if its monodromy with respect to any framing of the surface (see Section 4.3) acts trivially on the corresponding negative or dual cyclic homology class. Note that a direct variation on this result for arbitrary perverse schobers does not hold, as follows from a variant of Example 4.35. Theorem 5.8 generalizes the construction of relative Calabi–Yau structures on the topological Fukaya categories of framed surfaces of [2].
Theorem 5.8.
Let ℱ:Exit(G)→LinCatRdual be a G-parametrized perverse schober without singularities. Fix an edge e of G, and let 𝒩=ℱ(e) be the generic stalk of ℱ.
Suppose that𝒩is smooth and admits a left(n−1)-Calabi–Yau structure
η:R[n−1]→HH(𝒩)S1.
Suppose that the local system (see Remark 4.32)
HH(ℒℱ)S1:π1(ΣG)⟶π0AutRModR(HH(𝒩)S1)
preservesη. Then the functor
∂ℱ:∏e′∈G1∂ℱ(e′)⟶Γ(G,ℱ)
admits a leftn-Calabi–Yau structure.
Suppose that𝒩is proper and admits a right(n−1)-Calabi–Yau structure
η:R[n−1]→HH(𝒩)S1∗.
Suppose that the local system
HH(ℒℱ)S1∗:π1(ΣG)⟶π0AutRModR(HH(𝒩)S1∗)
preservesη. Then the functor
∏e′∈G1∂eve′:Γdual(G,ℱ)⟶∏e′∈G1∂ℱ(e′)
admits a rightn-Calabi–Yau structure.
Proof.
We only prove part (i); part (ii) is analogous. Using Proposition 4.17 and Lemma 4.33, we may assume that G has a single vertex v. We choose a framing ξ on ΣG. Let m be the valency of v. We choose a total order of the halfedges incident to v. Applying Proposition 5.2 to the spherical adjunction F=0𝒩:0↔𝒩:G, with the left n-Calabi–Yau structure on G arising from η, yields a left n-Calabi–Yau structure on R0𝒩m, which restricts on 𝒩×m to η×m. The diagram R0𝒩m gives rise to a perverse schober 𝒢v′ on the m-spider Gm, assigning to the incidence of the i-th halfedge with v the functor ϱi[−i].
Consider an internal edge h of G, which is by assumption a loop. The loop consists of two halfedges which lie in positions 1≤i<j≤m, and we orient h so that it first traces along the i-th halfedge and then the j-th halfedge. We modify 𝒢v′ by composing the functor 𝒢v′(v→𝑗h) with the autoequivalence ξ∗ℒℱ(h)[j−i−W(h)]:𝒩→𝒩.
We do this for each such internal edge h and denote the arising perverse schober on the m-spider by 𝒢v. We let 𝒢 be the G-parametrized perverse schober which restricts along Exit(Gm)→Exit(G) to 𝒢v. We have defined 𝒢 such that ℱ and 𝒢 have equivalent monodromy local systems. It follows from Proposition 4.34 that ℱ≃𝒢.
Using the above relative Calabi–Yau structure on 𝒢v′, that the monodromy of ℱ acts trivially on η, that the winding numbers of a framing are all even and that HH([1])=−idHH(𝒩), we find that ℱ satisfies the assumptions of Theorem 5.7. It follows that functor (5.5) admits the desired left n-Calabi–Yau structure.∎
Weak right Calabi–Yau structures on spherical functors
Consider a dualizable R-linear functor F:𝒟→𝒞 between proper R-linear ∞-categories with right adjoint G. If F admits a weak right Calabi–Yau structure, the arising fiber and cofiber sequence
id𝒟→uGF⟶id𝒟∗[1−n]
exhibits the shifted Serre functor id𝒟∗[1−n] as the twist functor of the adjunction F⊣G. If 𝒟 is smooth, then id𝒟∗ is an equivalence. To check that the adjunction F⊣G is spherical, it thus suffices to show that the cotwist functor is also an equivalence or alternatively that the unit of the adjunction F⊣G commutes with id𝒟∗ and that G admits a right adjoint H such that Im(F)≃Im(H) (see [7, Prop. 4.5]).
Conversely, suppose that F⊣G is a spherical adjunction, satisfying that the twist functor T𝒟 is equivalent to id𝒟∗[1−n]. The unit and counit maps of spherical adjunctions exhibit a rather special behavior: in the fiber and cofiber sequence
id𝒟→uGF→cu′T𝒟,
the map cu′ is a counit map of the adjunction E⊣F composed with T𝒟, up to composition with an autoequivalence GF≃T𝒟∘EF. By Lemma 2.34, this fiber and cofiber sequence looks very similar to diagram (3.2) appearing in the definition of a weak relative right Calabi–Yau structure on F. It is thus natural to ask whether F already admits a weak right n-Calabi–Yau structure. In this section, we prove that F can indeed be equipped with a weak right n-Calabi–Yau structure, under the assumption that 𝒞 is weak right (n−1)-Calabi–Yau and 𝒞,𝒟 are compactly generated (see Proposition 5.9).
The proof is rather indirect and relies on first lifting the spherical adjunction F⊣G to a perverse schober on the 3-spider, which might be thought of as a kind of resolution, as it renders trivial certain commutativity problems of diagrams involved in checking the existence of the relative Calabi–Yau structure. We then use an explicit description of the Serre functor on the global sections of this perverse schober that is only available in the proper setting. We get back to the original spherical adjunction by gluing with the Calabi–Yau structure of the zero functor 𝒞×2→0. We leave it as an interesting problem to find an alternative argument which applies in the smooth setting.
Proposition 5.9.
Let F:𝒟↔𝒞:G be a spherical adjunction of compactly generated, proper R-linear ∞-categories. Let T𝒟 be the twist functor of F⊣G. If there exists an equivalence T𝒟≃id𝒟∗[1−n] and a weak right (n−1)-Calabi–Yau structure on 𝒞, then F admits a weak right n-Calabi–Yau structure.
We will apply Proposition 5.9 to examples in Section 6.3.
Lemma 5.10.
Let F:𝒟↔𝒞:G be a spherical adjunction of compactly generated, proper R-linear ∞-categories. Let T𝒟 be the twist functor of F⊣G. Suppose that there is an equivalence T𝒟≃id𝒟∗[1−n] and that 𝒞 admits a weak right (n−1)-Calabi–Yau structure. Consider the spherical adjunction (see Proposition 5.1)
The functor SL admits a weak right n-Calabi–Yau structure.
Proof.
We show in Lemma 5.11 that the Serre functor id𝒱F3∗ is equivalent to a suspension of the twist functor T𝒱F3 of SL⊣S. The definition of T𝒱F3 thus gives us a fiber and cofiber sequence of endofunctors of 𝒱F3:
id𝒱f3→uSSL→𝜂id𝒱F3∗[1−n].
By assumption, 𝒞 admits a weak right (n−1)-Calabi–Yau structure, corresponding to an equivalence α:id𝒞≃id𝒞∗[1−n]. This gives us an equivalence
Fiber and cofiber sequence (5.7) gives rise to a commutative diagram in LinR(𝒱F3,𝒱F3)
with horizontal fiber and cofiber sequences and vertical equivalences. The map η is up to composition with an equivalence S∗(id𝒞×3)≃id𝒱F3∗SLLSL[1−n] given by a counit map of the adjunction SLL⊣SL, which can be seen as follows. By [16, Cor. 2.5.16], there exists an equivalence e:id𝒱F3∗[1−n]SLL≃T𝒱F3SLL≃S. By [7, Lem. 2.10], the composite natural transformation η∘eSL evaluates at each object of 𝒱F×3 to a counit map of SLL⊣SL so that η∘eSL is adjoint to a pointwise autoequivalence id𝒱F3∗SLL[1−n]≃id𝒱F3∗SLL[1−n]. This implies that η∘eSL is already a counit composed with id𝒱F3∗[1−n]. By Lemma 2.34, this shows that the natural transformation ν agrees with the counit cu~ from Construction 2.31, up to composition with an autoequivalence β of S∗(id𝒞×3∗)[1−n].
As we show next, the adjunction SL⊣S has the special feature that the map
is an equivalence. We denote the inverse image of β under this map by β′. We have already seen this in the special case that 𝒟=0 and 𝒞=RModR in the proof of Lemma 5.6. The argument here is very analogous: the functor SLS splits as SLS≃id𝒞×3⊕P with P the cotwist functor of SL⊣S, which permutes the three factors of 𝒞×3 cyclically by one step and then acts on each component as suspension or the cotwist functor of F⊣G. We thus have Mor(P,id𝒞×3)≃0 and Mor(P2,id𝒞×3)≃0 and
We adapt the choice of Calabi–Yau structure on 𝒞 by postcomposing α:id𝒞≃id𝒞∗[1−n] with β′. Note that ν∘S∗((β′)×3) is by construction equivalent to cu~. The existence of diagram (5.8) thus induces a relative dual Hochschild homology class σ∈HH(𝒱F3,𝒞×3)∗ which defines a weak right n-Calabi–Yau structure on SL. We finally remark that σ restricts at 𝒞×3 to the class corresponding to (β′∘α)×3.∎
Lemma 5.11.
Under the assumptions of Lemma5.10, the shifted twist functor T𝒱F3[n−1] of adjunction (5.6) is equivalent to the Serre functor id𝒱F3∗.
Proof.
In this proof, we will use the following simplified and abusive notation for elements of 𝒱F3: given an element of 𝒱F3, that is, a diagram d→c1→c2, with the morphism d→c1 lying in the Grothendieck construction of F, meaning that it encodes a morphism F(d)→c1, we simply write it as a tuple (d,c1,c2). We similarly write elements c1→c2 of Fun(Δ1,𝒞) as pairs (c1,c2) and elements d→c of 𝒱F2≃fib(ϱ1)⊂𝒱F3 as pairs (d,c). Given elements x=(c1,c2)∈Fun(Δ1,𝒞) and d∈𝒟, we will also write (d,x) for (d,c1,c2).
The Serre functor of 𝒞 is U𝒞≃[n−1], the Serre functor of 𝒟 is given by T𝒟[n−1] and the Serre functor of Fun(Δ1,𝒞) is denoted by U2. The functor U2 is given by the tensor product of the Serre functors of Fun(Δ1,RModR) and 𝒞. By Lemma 5.5 and Remark 5.4, U2 is thus given by the assignment
U2:(c1,c2)↦(c2,cof(c1→c2))[n−1].
A straightforward computation shows that the twist functor T𝒱F3:𝒱F3→𝒱F3 of SL⊣S is given by the assignment
This assignment is to be understood as in Remark 5.4, meaning the apparent functor corresponding to the above formulas constructed using the universal properties of the involved lax limits.
Consider the relative suspension functor τ:𝒱F2→𝒱F2 of [16, Def. 2.5.8], which is on objects given by mapping (d,c) to (cof(d→G(c)),cof(F(d)→c)). The functor τ is an equivalence by the sphericalness of F and [16, Cor. 2.5.10], and this implies that
This shows that T𝒱F3[n−1] is a Serre functor and thus equivalent to id𝒱F3∗.∎
Proof of Proposition <xref rid="j_JNCG663_stat_005_139" ref-type="statement">5.9</xref>.
Consider the functor SL:𝒱F3→𝒞×3 from Lemma 5.10 which admits a weak right n-Calabi–Yau structure. Applying Theorem 3.16 to the pullback diagram in LinCatR
yields the desired weak right n-Calabi–Yau structure on F.∎
Examples
We begin in Section 6.1 by describing Fukaya–Seidel categories as the global sections of perverse schobers on the disc and using this to construct relative Calabi–Yau structures on these. In Section 6.2, we describe a special case of Theorem 5.8 concerning relative Calabi–Yau structures on periodic topological Fukaya categories of marked surfaces. Finally, we observe in Section 6.3 that the derived categories of relative Ginzburg algebras of n-angulated surfaces admit relative left n-Calabi–Yau structures and further exhibit in some cases weak right n-Calabi–Yau structures on the R-linear versions of the finite derived categories of these relative Ginzburg algebras, where R is an arbitrary base 𝔼∞-ring spectrum.
Fukaya–Seidel categories
The cosheaves of partially wrapped Fukaya categories of [20] give rise to perverse schobers. We will explain this in this section in the setting of Lefschetz fibrations over the disc. We remark that a related construction of Fukaya–Seidel categories using perverse schobers appears in [27].
We work in the setup of partially wrapped Fukaya categories of [20]. The original construction of Fukaya–Seidel categories in the different setup of [37], as a directed A∞-category, can be treated in a similar way using perverse schobers.
Let ℂ≥0 be the half-plane (considered as a Liouville sector). Let π:X→ℂ≥0 be a Lefschetz fibration (in the sense of [20]), with X a Liouville sector, with regular Weinstein fiber F and core 𝔣⊂F. Let 2n be the dimension of M. We assume that 2n≥4. We further assume that the wrapped Fukaya category 𝒲(F) of the fiber is weakly left (n−1)-Calabi–Yau, which is shown under minor assumption on F in [19].
The wrapped Fukaya category 𝒲(X) is equivalent to the partially wrapped Fukaya category 𝒲(X¯,𝔣) of the Liouville manifold X¯ arising from X with a stop at 𝔣. This A∞-category is called the Fukaya–Seidel category of the Lefschetz fibration π. We will denote it by FS(π)≔𝒲(X).
From FS(π), we obtain a k-linear stable ∞-category 𝒟(FS(π))∈LinCatk by first choosing a quasi-equivalent dg category to FS(π) and then passing to its derived ∞-category.
As described in [20, Exm. 1.31], the Fukaya–Seidel category FS(π) arises as the homotopy colimit of a diagram of A∞-categories with values given by the A∞-categories 𝒲(F)⊗A2,𝒲(F) and Perf(k). This diagram describes a perverse schober and is described in more detail in Construction 6.3. This allows us to obtain a weak relative left n-Calabi–Yau structure on 𝒟(FS(π)).
Theorem 6.1.
Let π:X→ℂ≥0 be a Lefschetz fibration as above.
The derived∞-category of the Fukaya–Seidel category𝒟(FS(π))arises as the∞-category of global sections of the perverse schoberℱonℂ≥0from Construction 6.3with singularities at the singular values ofπand generic stalk𝒟(𝒲(F)).
Passing to derived∞-categories, the canonical functor
𝒲(F)→FS(π)
known as the cup/Orlov functor
See [42]. The left adjoint is called the cap functor.
agrees with the spherical boundary corestriction functor∂ℱ(see equation (4.4)). The functor∂ℱadmits a weak leftn-Calabi–Yau structure, exhibiting𝒟(FS(π))as weak relative leftn-Calabi–Yau.
Remark 6.2.
The derived Fukaya–Seidel category FS(π)∈LinCatk is smooth being the colimit of smooth ∞-categories and proper as it is generated by the thimbles. There is a pushout diagram in LinCatk of the following form (see [20, Thm. 1.20]).
Thus, FS(π) can be seen as a smooth and proper resolution of the smooth 𝒟(𝒲(X¯)). The Serre functor on 𝒟(FS(π)) is given by part (ii) of Theorem 6.1 by a shift of the cotwist functor of the spherical adjunction ∂ℱ↔radj(∂ℱ). Note that spherical functors commute with their (co)twist functors (see [7, Lem. 2.2]). The Serre functor of FS(π) thus stabilizes the stable subcategory generated by the image of ∂ℱ.
Furthermore, by gluing with the Calabi–Yau functor 𝒟(𝒲(F))→0 via Theorem 3.15, we find that 𝒟(𝒲(X¯)) inherits a weak left n-Calabi–Yau structure, recovering the result of [19].
Construction 6.3.
Let n be the number of singular values of the Lefschetz fibration π. Consider the following ribbon graph Gπ.
We can embed Gπ into ℂ≥0, making it into a spanning ribbon graph, such that each vertex si lies at a singular value of π and the external edge incident to v1 ends on the unique boundary component of ℂ≥0. The embedding of Gπ into ℂ≥0 decomposes ℂ≥0 into A2-sectors, lying near the vertices v1,…,vn, not containing any singular values, as well as n half-planes containing the singular values si. The derived wrapped Fukaya categories of the inverse images of the half-planes are each equivalent to 𝒟(k). The Fukaya–Seidel category arises by [20, Example 1.31] as the homotopy colimit of a diagram of A∞-categories, indexed by the opposite of the exit path category of Gπ. This diagram assigns
to each edge of Gπ an A∞-category Morita equivalent to the wrapped Fukaya category 𝒲(F) of the fiber,
to each vertex vi an A∞-category Morita equivalent to 𝒲(F)⊗A2, and
to each vertex si an A∞-category Morita equivalent to Perf(k).
Passing to derived ∞-categories and right adjoint functors yields a diagram ℱ:Exit(G)→LinCatk, which is readily verified to describe a perverse schober with singularities at s1,…,sn. The spherical adjunctions at the vertices si arise from the spherical objects in 𝒲(F) given by the vanishing cycles of the Lefschetz fibration.
Note that we can contract the ribbon graph Gπ to the following ribbon graph Gπ′.
This allows us to understand the ∞-category of global sections of ℱ as a non-full subcategory of 𝒟(𝒲(F)⊗An)≃Fun(Δn−1,𝒟(𝒲(F))), similar to the directed subcategory construction of [37]. From this perspective, the thimbles of the Lefschetz fibration amount to coCartesian sections of the Gπ′-parametrized perverse schober of the following form: for Xi∈𝒟(𝒲(F)), the i-th vanishing cycle, the corresponding thimble Yi is given as follows:
satisfying that the restriction of Yi to the j-th internal edge ej is given by Xi if i=j and 0 if i≠j.
We next briefly discuss spherical objects and then show that these give rise to functors with Calabi–Yau structures.
We let k be a commutative ring and 𝒞 a dualizable k-linear ∞-category. We fix an object X∈𝒞c whose endomorphism object is equivalent to the singular complex of the (n−1)-sphere for some n≥2, meaning that
Map𝒞(X,X)≃k⊕k[−n+1]∈𝒟(k).
The object X gives rise to a k-linear adjunction
-⊗kX:𝒟(k)⟷𝒞:Map𝒞(X,-).
By the assumption on X, the twist functor T𝒟(k)≃cof(id𝒟(k)→-⊗kMap𝒞(X,X)) is equivalent to the (n−1)-fold loop functor [−n+1] and thus an equivalence. We call the object X an (n−1)-spherical object if the cotwist functor T𝒞 is also an equivalence and adjunction (6.1) is thus a spherical adjunction. In this case, the right adjoint of Map𝒞(X,-) is given by -⊗kX[n−1]. If 𝒞 is proper and compactly generated, with Serre functor U, the adjunction is spherical if and only if U(X)≃X[n−1] so that we specialize to the usual notion of a spherical object (see, for instance, [24, Def. 8.1]): one implication of this can be proven using [7, Prop. 4.5] and the fact that iterated adjoints between proper, compactly generated ∞-categories are obtained by compositions with powers of the Serre functors.
Lemma 6.4.
Let n≠1 and X∈𝒞 an (n−1)-spherical object in a dualizable k-linear ∞-category.
If𝒞is proper and admits a weak right(n−1)-Calabi–Yau structure, then the functor
-⊗kX:𝒟(k)⟶𝒞
admits a compatible weak rightn-Calabi–Yau structure.
If𝒞is smooth and admits a weak leftn-Calabi–Yau structure, then the functor
Map𝒞(X,-):𝒞⟶𝒟(k)
admits a compatible weak leftn-Calabi–Yau structure.
Proof.
We only show part (1); part (2) can be shown analogously. Let σ:k[n−1]→HH(𝒞)∗ be a weak right (n−1)-Calabi–Yau structure on 𝒞. The class σ gives rise to the following diagram.
Note that id𝒟(k)∗≃id𝒟(k). Since there are no natural transformations from id𝒟(k) to id𝒟(k)[−n+1], the above diagram is equivalent to the following diagram.
This diagram clearly admits a null-homotopy which defines a non-degenerate relative dual Hochschild class k[n]→HH(𝒟(k),𝒞)∗, restricting to σ on HH(𝒞)∗, thus exhibiting the desired weak right (n+1)-Calabi–Yau structure on the functor -⊗kX:𝒟(k)→𝒞.∎
Proof of Theorem <xref rid="j_JNCG663_stat_006_145" ref-type="statement">6.1</xref>.
We begin with showing part (i). By the cosheaf properties of partially wrapped Fukaya categories (see [20, Exm. 1.31]), we find that FS(π) arises as the homotopy colimit of the diagram of A∞-categories indexed by Exit(Gπ)op described in Construction 6.3. It remains to show that the passage to the derived ∞-category turns the A∞-categorical homotopy colimit
We point out that in contrast to the category of dg categories, the category of A∞-categories does not admit a suitable model structure so that the notion of ‘homotopy colimit’ employed in [20] is not understood in a model categorical sense. One can nevertheless expect that the passage to the derived ∞-categories always turns such homotopy colimits into ∞-categorical colimits.
into an ∞-categorical colimit. This follows from two observations. Firstly, by the universal property of the colimit, there is a comparison functor ℋ(Gπ,ℱ)→𝒟(FS(π)). Secondly, both stable ∞-categories are generated by the thimbles, and on these, the above functor is a quasi-equivalence.
To obtain the weak left Calabi–Yau structure in part (ii), we combine Theorem 3.15, Lemma 6.4, as well as Proposition 5.2 applied to the spherical adjunction 𝒟(𝒲(F))↔0. The existence of the weak right Calabi–Yau structure similarly follows using the observation that the limit of ℱ is equivalent to the limit of a diagram of proper subcategories in LinCatkdual.∎
Periodic topological Fukaya categories
Let k be a field. We fix an integer n≥1 and denote by k[tn±] the graded ring of Laurent polynomials in a formal variable tn in degree n. Note that if n is even, then k[tn±] is graded commutative. A k[tn±]-module amounts to an n-periodic k-linear chain complex. If n is even, we thus refer to k[tn±]-linear ∞-categories as n-periodic k-linear ∞-categories. In the following, we discuss how Theorem 5.8 specializes to periodic topological Fukaya categories.
If n is even, we set m=n. Otherwise, we set m=2n. As explained in [14], given an oriented marked surface S if m=2, or more generally a marked m-spin surface, with spin structure denoted by ξ, one can associate a topological Fukaya of S with values in the n-periodic stable ∞-category 𝒟(k[tn±]). The analogue of its construction in [14] in terms of the global sections of perverse schobers is as follows: there is a unique (up to equivalence) k[tm±]-linear locally constant perverse schober ℱ on S with generic stalk 𝒟(k[tn±]) and whose monodromy local system with respect to ξ, in the sense of Remark 4.31, is trivial. We call its global sections the n-periodic topological Fukaya category of S and denote it by Fuk(S,𝒟(k[tn±])). The corresponding ∞-category of locally compact global sections of ℱ is by Lemma 4.18 equivalent to IndFuk(S,𝒟(k[tn±]))fin.
Theorem 6.5.
Let k be a field with char(k)≠2 and m,n as above.
Thek[tm±]-linear topological Fukaya categoryFuk(S,𝒟(k[tn±]))valued in the derived∞-category𝒟(k[tn±])ofn-periodic chain complexes admits a relative left(n+1)-Calabi–Yau structure.
TheInd-finite subcategoryIndFuk(S,𝒟(k[tn±]))finis proper and admits a relative right(n+1)-Calabi–Yau structure. Further, if each boundary component of has at least one marked point, then S Fuk(S,𝒟(k[tn±]))≃IndFuk(S,𝒟(k[tn±]))fin.
Proof.
Combine Theorem 5.8, Lemma 6.6 for n odd and Remark 4.19.∎
Lemma 6.6.
Let k be a field with char(k)≠2. Let n≥1 be odd. Then 𝒟(k[tn±]) is smooth and proper as a k[t2n±]-linear ∞-category and further admits left and right n-Calabi–Yau structures.
Proof.
Denote A≔k[tn±]. The k[t2n±]-linear enveloping algebra Ae of A is given by the graded commutative dg algebra k[t±,s±]/(s2−t2), with generators t,s in degrees n and satisfying st=(−1)m2ts=−ts (graded commutativity), as well as s2=t2. As a right Ae-module, A is equipped with the action 1.t=t and 1.s=−t. As a left (Ae)op-module, A is equipped with the action t.1=−t and s.1=t. We denote by A¯ the right Ae-module A with the action 1.t=t and 1.s=t.
We consider Ae as a right module over itself. There is a retract of right Ae-modules
A→1↦1−s−1tAe→1↦1A
since the composite is given by multiplication by 1−1.s−1t=1+t−1t=2≠0 and thus invertible. There is a similar retract
A¯→1↦1+s−1tAe→1↦1A¯
and Ae≃A⊕A¯.
It follows that A is compact as a right Ae-module. The inverse dualizing functor id𝒟(A)! is given by the tensor product with the left Ae-module A!=RHomAe(A,Ae). One finds RHomAe(A,A¯)≃RHomAe(A¯,A)≃0. We thus have RHomAe(A,Ae)≃RHomAe(A,A)≃RHomAe(Ae,A)≃A on k-linear homology, with 1∈A being the image of ϕ:A→Ae, 1↦1−s−1t. The element t∈A corresponds to ϕ′:1↦t−s. The left action of Ae on HomAe(A,Ae)≃A is determined by t.ϕ=ϕ′ and s.ϕ=−ϕ′. It follows that A!=RHomAe(A,Ae)≃A[−n] as left Ae-modules, since the shift by n preserves the homology of A over k but flips the signs of the actions of t and s (since n is odd). This shows that id𝒟(A)!≃id𝒟(A)[−n], as desired. Composing with id𝒟(A)∗ also yields id𝒟(A)≃id𝒟(A)∗[−n].
It remains to show that the (dual) Hochschild homology classes of these weak left and right m-Calabi–Yau structures lift to negative cyclic homology and dual cyclic homology, respectively. We show this by proving the triviality of the S1-action on HH(𝒟(A)). We have HH(𝒟(A))≃RHomAe(A!,A)≃RHomAe(A,A)[n]≃A[n]≃A. The S1-action on the k[t2n±]-linear Hochschild homology HH(𝒟(k[t2n±]))≃k[t2n±] is trivial by Remark 2.27 and the same thus holds for its image k[t2n±]⊂A under
HH(-⊗k[t2n±]A):HH(𝒟(k[t2n±]))→HH(𝒟(A)).
We are left with determining the S1-action on the summand k[t2n±][n]⊂A≃k[t2n±]⊕k[t2n±][n]. An S1-action on a k[t2n±]-module is the same as a k[t2n±][S1]-module structure, where k[t2n±][S1]≃k[t2n±]⊗kk[S1]≃k[t2n±]⊗kk[s1]/(s12=0), with |s1|=1 (see, for instance, [21, Prop. 3.3] for the latter equivalence). Since 2n≥2, the action of s1, and thus of S1, on k[t2n±][n] is trivial for degree reasons, concluding the proof.∎
Remark 6.7.
There is a 2-periodic version of the sphere spectrum, and topological Fukaya categories with coefficients in the modules over this ring spectrum have been considered in [33]. Interestingly, the 2-periodic sphere spectrum is not an 𝔼∞-ring spectrum but only an 𝔼2-ring spectrum so that Theorem 6.5 cannot be directly lifted to this setting.
Relative Ginzburg algebras of surfaces
Fix a base 𝔼∞-ring spectrum R, and let n≥3. Let S be a marked surface, equipped with an n-valent spanning ribbon graph G. It is dual to a so-called ideal n-angulation, roughly meaning a decomposition of S into n-gons with vertices at the marked points of S. There is an associated perverse schober ℱG(R) (see [6]). If R is discrete, that is, a commutative ring, the ∞-category of global sections Γ(G,ℱG) is equivalent to the derived ∞-category 𝒟(𝒢G) of a relative Ginzburg algebra 𝒢G (see [6]). The generic stalk of ℱG is given by the ∞-category Fun(Sn−1,RModR) of RModR-valued local systems on the (n−1)-sphere. At every vertex of G, the spherical adjunction underlying the perverse schober ℱG(R) is given by the adjunction
f∗:RModR⟷Fun(Sn−1,RModR):f∗
arising from the pullback functor along the inclusion of the boundary f:Sn−1→Dn≃∗ of the n-ball.
As shown in [7], there is an equivalence of R-linear ∞-categories
RModR[tn−2]≃Fun(Sn−1,RModR),
where R[tn−2] denotes the free R-linear algebra generated by R[n−2]∈RModR. Under this equivalence, the functor f∗ is identified with the pullback functor ϕ∗ along the morphism R[tn−2]→R determined by mapping tn2 to 0. Note that if R=k is a field, then k[tn−2] is the graded polynomial algebra with generator in degree |tn−2|=n−2.
If R=k is a field, it is shown in [2, Thm. 5.7] that the functor f∗ admits a relative left n-CY structure, which further restricts to a left (n−1)-Calabi–Yau structure on Fun(Sn−1,RModk). The next theorem states that Theorem 5.7 applies to give a relative Calabi–Yau structure on the global sections of ℱG(k) if either n is odd or the spanning graph G is orientable in the following sense.
Definition 6.8.
Let n be even. We call the n-valent spanning graph G of S orientable if there exist choices of orientations of the edges of G such that the directions of the halfedges at any vertex of G alternate in their cyclic order.
Theorem 6.9.
Let G be an n-valent spanning graph of a marked surface S and 𝒟(𝒢G) the derived ∞-category of the corresponding relative Ginzburg algebra 𝒢G. If n is odd or G orientable, then the functor
For the construction of ℱG(k), each vertex v of G is equipped with a choice of total order of its incident halfedges, whose corresponding edges we denote by e1,…,en. The functor
ℱG(k)(v→ei):𝒱f∗n→Fun(Sn−1,RModk)
is either given by ϱi or φ∗∘ϱi, with φ∗ the autoequivalence of Fun(Sn−1,RModk)≃𝒟(k[tn−2]) given by pullback along
φ:k[tn−2]→tn−2↦(−1)ntn−2k[tn−2].
Inspecting Proposition 5.2, using HH([1])S1=−id, and ignoring the equivalence φ∗ for the moment, we see that the signs alternate cyclically of the classes describing the left (n−1)-Calabi–Yau structures of ℱG(k)(ei), arising from restricting the relative left n-Calabi–Yau structure of ℱG(k)(v). If n is odd, the map HH(φ∗)S1 reverses the sign of the class, whereas if n is even, φ∗=id𝒟(k[tn−2]) fixes the class.
We suppose that n is even and choose an orientation of G. Choose further for each vertex v of G an incident halfedge; if it points outward from v, we equip ℱG(k)(v) with the relative Calabi–Yau structure from Proposition 5.2, and if v points inward, we equip ℱG(k)(v) with the same relative Calabi–Yau structure, except for reversing the sign of the relative dual cyclic homology class. With these choices, Theorem 5.7 applies.
It remains to consider the case that n is odd. For all vertices v, we equip ℱG(k)(v) with the relative Calabi–Yau structure from Proposition 5.2. Consider the two vertices v,v′ incident to an edge ei in position i in the total order of halfedges at v and in the i′-th position in the total order of halfedges at v′. The equivalence φ∗ appears in one of the functors ℱG(k)(v→ei), ℱG(k)(v′→ei) if and only if the difference i−i′ of the two positions in the total orders of the two halfedges of ei is even. We thus see that the induced Calabi–Yau structures on ℱG(k)(ei) are compatible for any edge ei so that Theorem 5.7 again applies.∎
We next note a variant of Theorem 6.9 on the existence of a weak right n-Calabi–Yau structure on the proper R-linear ∞-category Γdual(G,ℱG(R)), with n even and R an arbitrary 𝔼∞-ring spectrum.
Theorem 6.10.
Suppose that n is even and G orientable. Then the functor
∏e∈G1∂eve:Γdual(G,ℱG(R))⟶∏e∈(G)1∂ℱG(R)(e)
admits an R-linear weak right n-Calabi–Yau structure.
Proof.
The proof of Theorem 6.9 directly translates by using Lemma 6.11.∎
The adjunction f∗⊣f∗ restricts to the adjunction of proper R-linear ∞-categories:
f¯∗:RModR⟷IndFun(Sn−1,RModRperf):f¯∗.
We note that
Fun(Sn−1,RModRperf)≃Fun(Sn−1,RModR)fin.
Lemma 6.11.
For any n≥1, the R-linear ∞-category IndFun(Sn−1,RModRperf) admits a weak right (n−1)-Calabi–Yau structure and the functor f¯∗ admits a compatible weak right n-Calabi–Yau structure.
Proof.
The adjunction f∗⊣f∗ is spherical with twist functor TRModR≃[1−n] (see [7]); the same thus holds for f¯∗⊣f¯∗. By Proposition 5.9, the functor f¯∗ thus admits a weak right n-Calabi–Yau structure if IndFun(Sn−1,RModRperf) admits a weak right n-Calabi–Yau structure.
Applying Fun(-,RModRperf) to the following pushout diagram of spaces
and Ind-completing, we obtain the pullback diagram of compactly generated R-linear ∞-categories.
Applying Theorem 3.16 to (6.2), it now follows by induction on i that the R-linear ∞-category IndFun(Si,RModRperf) admits a weak right i-Calabi–Yau structure and that f¯∗:RModR→IndFun(Si,RModRperf) admits a weak right (i+1)-Calabi–Yau structure, concluding the proof.∎
Acknowledgements
The present paper is based on and refines results from the author’s Ph.D. thesis, advised by Tobias Dyckerhoff. I thank him for helpful discussions and feedback. I thank Gustavo Jasso for pointing out a gap in the proof of Lemma 2.4 in a previous version of the article. I further thank Chris Brav, Fabian Haiden and Bernhard Keller for helpful discussions.
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