Ginzburg algebras of triangulated surfaces and perverse schobers
aa r X i v : . [ m a t h . A T ] J a n Ginzburg algebras of triangulated surfaces and perverse schobers
Merlin ChristJanuary 7, 2021
Abstract
Ginzburg algebras associated to triangulated surfaces provide a means to categorify thecluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of sucha Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetzfibration over the surface. Inspired by this perspective we provide a description of the full derivedcategory in terms of a perverse schober. The main novelty is a gluing formalism describing theGinzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a firstapplication we give a new proof of the derived invariance of these Ginzburg algebras under flipsof an edge of the triangulation. Finally, we note that the perverse schober as well as the resultinggluing construction can also be defined over the sphere spectrum.
Contents ∞ -categories of ∞ -categories . . . . . . . . . . . . . . . . . 82.2 Modules over ring spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Differential graded categories and their modules . . . . . . . . . . . . . . . . . . . 122.4 A model for the derived ∞ -category of a dg-algebra . . . . . . . . . . . . . . . . 132.5 Morita theory of dg-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Semiorthogonal decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 V nf ∗ V f ∗ and V f ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Gluing Ginzburg algebras via perverse schobers 50
Cluster algebras were introduced by Fomin and Zelevinsky [FZ02] as a class of commutativealgebras equipped with a combinatorial structure relating different subsets of the algebra calledclusters. Since then, there has been a great interest in cluster algebras and their relation to othersubjects including Teichmüller theory, polyhedral surfaces, representation theory of quivers andaspects of noncommutative geometry such as Calabi-Yau algebras, Calabi-Yau categories andstability conditions. A survey with many references can be found in [Kel08], we also refer to thecluster algebra portal [Fom] for further surveys and information regarding cluster algebras.Relevant for this work is a particular class of cluster algebras associated to oriented markedsurfaces equipped with an ideal triangulation introduced in [GSV05, FG06b, FG09], and furtherstudied in [FST08, FT18]. These cluster algebras can be described in two different ways. Thefirst perspective is geometric and provides a description in terms of the decorated Teichmüllerspaces of the surfaces. The cluster variables arise as lambda lengths, which form the coordinatesof the Teichmüller space. These lambda lengths satisfy an analogue of the classical Ptolemyrelations, which gives rise to the cluster exchange relations. The second perspective makesdirect use of the combinatorics of the ideal triangulation. The mutation matrix used to definethe cluster algebra arises as the signed adjacency matrix of the ideal triangulation, which countsthe number of incidences of the ideal triangles. In most cases, the resulting algebra does notdepend on the choice of ideal triangulation but only on the underlying marked surface.This second perspective shows that cluster algebras of surfaces can be considered as clus-ter algebras associated to quivers, which can be categorified via 2-Calabi-Yau (CY) triangu-lated categories, called cluster categories, and 3-CY triangulated categories. To describe the3-CY categorification of the cluster algebra associated to a quiver Q , one chooses a non-degenerate potential W . The 3-CY categorification is then given by the derived category ofthe Ginzburg algebra G ( Q, W ) associated to the quiver with potential (
Q, W ). The 2-CY clus-ter category can be obtained from the derived category of the Ginzburg algebra via the Verdierquotient D ( G ( Q, W )) perf / D ( G ( Q, W )) fin , see [Ami09]. There is also a direct relation betweenthe Ginzburg algebras and the combinatorics of the cluster algebras, we refer to [Kel12] for asurvey.To describe the results of this work, we describe the construction of the quiver Q ◦ T , anda choice of non-degenerate potential W T , associated to an ideal triangulation T , see [LF09,GLFS16]. We assume for simplicity that T has no self-folded triangles. We define the quiver Q ◦ T with vertices the interior edges of T and an arrow a : i → j for each ideal triangle containingthe edges i, j where the edge j follows the edge i in the clockwise order of the edges of theideal triangle induced by the orientation of the surface. The non-degenerate potential W T = W ′ T + W ′′ T ∈ kQ ◦ T consists of a part W ′ T which is the sum of the clockwise 3-cycles inscribed inthe interior ideal triangles of T and a part W ′′ T which is a sum of counter-clockwise cycles, onefor each interior vertex of T .To above mentioned 2-CY and 3-CY categorifications can be described in terms of thecombinatorial geometry of T , see [QZ17] and references therein for the 2-CY cluster category nd [Qiu18,QZ14] for the finite part of the derived category of the Ginzburg algebra G ( Q ◦ T , W ′ T ).For us most relevant is the realization of the finite part of the derived category of G ( Q ◦ T , W ′ T ) asa full subcategory of the Fukaya category of a Calabi-Yau 3-fold Y ◦ equipped with a Lefschetzfibration π : Y ◦ → Σ := S \ ( M ∩ S ◦ ) to the open surface due to Smith [Smi15]. Inspired by thegeometry of π we give in this paper a description of the entire unbounded derived category ofthe Ginzburg algebra G ( Q ◦ T , W ′ T ) in terms of the global sections of a perverse schober.Before we describe our model for D ( G ( Q ◦ T , W ′ T )), we highlight the relation to a model for thepartially wrapped Fukaya categories of graded surfaces or equivalently the derived categoriesof gentle algebras [HKK17, LP20]. Consider an ideal triangulation of a graded marked surface S and the dual ribbon graph Γ. The Fukaya category of the surface S is equivalent to thedg-category of global sections of a constructible cosheaf of dg-categories on the ribbon graph Γ,see [DK15, HKK17]. The cosheaf description of the Fukaya category categorifies the statementthat the middle cohomology H Γ (Σ , Z [1]) of the open surface Σ with support on Γ is equivalentto the abelian group of global sections of a constructible cosheaf H Γ ( Z [1]) on Γ whose stalk ata point x is the homology H Γ ∩ U ( U, Z [1]) of a small neighbourhood x ∈ U ⊂ Σ with support onΓ ∩ U . Our model describes the derived category of the Ginzburg algebra in terms of the globalsections of a different constructible cosheaf of dg-categories on Γ. Denote by Γ ◦ the ribbongraph obtained by removing all exterior edges of Γ. Decategorified, the idea behind our modelis to express the middle cohomology of the 3-fold Y ◦ with support on π − (Γ ◦ ) in terms of theabelian group of global sections with support on Γ ◦ of the perverse pushforward π ∗ ( Z [3]) to Σ,which in turn is equivalent to the global sections with support on Γ ◦ of a constructible cosheafH Γ ( π ∗ Z [3]) on Γ. We will not provide a systematic categorification of the perverse pushforwardfunctor π ∗ , but rather provide an explicit description of the categorification of the constructiblecosheaf H Γ ( π ∗ Z [3]). This will be achieved by constructing a perverse schober on the surface thatis classified locally, at every critical value of Smith’s Lefschetz fibration, by the Ind-completeversion of the spherical adjunction W ( T ∗ S ) ←→ D ( k ) perf . The explicit computability of our model then arises from a concrete algebraic description ofthis adjunction, as well as the resulting categorification of H Γ ( π ∗ Z [3])) in terms of variants ofWaldhausen’s S • -construction. A full definition of the notion of a perverse schober on a surface isnot yet documented in the literature, we thus introduce a framework for the treatment of perverseschobers on surfaces which are parametrized by ribbon graphs. Our definition of parametrizedperverse schober can be seen as a generalization of the approach to topological Fukaya categoriesof surfaces of [DK18, DK15], allowing for the treatment of nonconstant coefficients. The mainresult of this paper is the following. Theorem 1.
Let T be an ideal triangulation of an oriented marked surface S and consider thedual ribbon graph Γ . There exists a Γ -parametrized perverse schober F T whose stable ∞ -categoryof global sections with support on Γ ◦ satisfies H Γ ◦ (Γ , F T ) ≃ D ( G ( Q ◦ T , W ′ T )) , i.e. is equivalent to the unbounded derived ∞ -category of the Ginzburg algebra G ( Q ◦ T , W ′ T ) . Note that if T contains no interior vertices, the potential W ′ T = W T is non-degenerate. Givenan ideal triangulation T with interior vertices, the potential W ′ T is in general degenerate. In thiscase, the Ginzburg algebra G ( Q ◦ T , W ′ T ) does not fully capture the cluster combinatorics.Informally, Theorem 1 can be summarized as the statement that the derived ∞ -category D ( G Q ◦ T , W ′ T )) arises via the gluing of simpler ∞ -categories. The pieces used in the gluingconstruction are the derived ∞ -categories of certain relative Ginzburg algebras of n -gons. Thisterminology was suggested by Bernhard Keller in his ICRA 2020 lecture series on relative Calabi-Yau structures. The derived ∞ -category of a relative Ginzburg algebra also appears as the -category of global sections H (Γ , F T ) of the parametrized perverse schober F T (without anyrestrictions on the support). The ∞ -category H (Γ , F T ) contains H Γ ◦ (Γ , F T ) ≃ D ( G ( Q ◦ T , W ′ T ) asa full subcategory. The passage from all global sections to global sections with support on theinterior thus constitutes a loss of information which explains why the non-relative Ginzburgalgebras cannot directly be glued. In terms of the underlying cluster algebras, our gluingconstruction seems to be a special case of the procedure of amalgamation and defrosting ofcluster algebras of [FG06a].To make the gluing construction of D ( G ( Q ◦ T , W ′ T )) work, we need to determine the correctway to glue the pieces. Making different choices would lead to different signs of the differentialsof the Ginzburg algebra. The total choice of signs is equivalent to a choice of spin structure onthe open surface S \ ( M ∩ S ◦ ), see Section 7.1.The formalism used for the description of the perverse schober F T works not only in the k -linear setting but also over the sphere spectrum. Many of our results naturally extend to thismore general setting, see Section 7.2.In Section 1.1, we recall the full definition of the 3-CY Ginzburg algebra and continue byintroducing relative Ginzburg algebras. In Section 1.2 contains a discussion of parametrizedperverse schobers and Smith’s results. In Section 1.3 we describe the gluing construction of theGinzburg algebra. A quiver Q consists of a finite set of vertices, denoted Q , and a finite set of arrows, denoted Q , together with source and target maps s, t : Q → Q . A quiver is called graded if eacharrow carries an integer labelling. Given a graded quiver Q , we denote by kQ the graded pathalgebra over a commutative ring k . A potential W for a quiver Q is an element of the cyclicpath algebra kQ cyc , meaning the algebra of k -linear sums of cyclic paths.For the definition of the Ginzburg algebra, due to [Gin06], we follow [Kel11]. Consider aquiver with potential ( Q, W ). We denote by Q ′ the graded quiver with the same set of verticesas Q and graded arrows of the following three kinds.• An edge a : i → j for each a : i → j ∈ Q of degree 0• An edge a ∗ : j → i for each a : i → j ∈ Q of degree 1.• An edge b i : i → i for each i ∈ Q of degree 2.The cyclic derivate ∂ a : kQ cyc → kQ with respect to a ∈ Q is the k -linear map taking a cycle c to ∂ a c = P c = uav uv , where u, v ∈ kQ are allowed to be lazy paths. We denote the lazy pathat a vertex i ∈ Q by p i . We define the Ginzburg algebra G ( Q, W ) to be the dg-algebra whoseunderlying graded algebra is given by the graded path algebra kQ ′ and whose differential d isdetermined by the following action on the generators. a a ∗ ∂ a Wb i X a ∈ Q p i [ a, a ∗ ] p i Note that G ( Q, W ) is not the completed Ginzburg algebra, as for example considered in [KY11,Smi15]. We will not consider completed Ginzburg algebras in this paper. In terms of theassociated derived ∞ -categories of these dg-algebras this does not mean much of a loss, becausethe derived ∞ -category of the completed Ginzburg algebra can be realized as a fully subcategoryof the derived ∞ -category of the non-completed Ginzburg algebra. This perspective howeverneglects the additional topological structure of the completed Ginzburg algebra, see for examplethe Appendix in [KY11]. e now introduce a relative version of the Ginzburg algebra G ( Q T , W ′ T ) associated to anideal triangulation T of an oriented marked surface S . We again assume for simplicity that T has no self-folded triangles, the more general case is discussed in Section 6.1. We define a quiver Q T by adapting the definition of the quiver Q ◦ T to include the boundary of S . We let Q T be thequiver with a vertex for each edge of T (including boundary edges) and an arrow a : i → j foreach ideal triangle containing the edges i, j where the edge j follows the edge i in the clockwiseorder. The quiver Q T contains a clockwise 3-cycle T ( f ) for each ideal triangle f of T . We definethe potential W ′ T = X f T ( f ) ∈ kQ T . We denote by ˜ Q T the graded quiver with the same set of vertices as Q T and graded arrows ofthe following three kinds.• An edge a : i → j of degree 0 for each a : i → j ∈ ( Q T ) .• An edge a ∗ : j → i of degree 1 for each a : i → j ∈ ( Q T ) .• An edge b i : i → i of degree 2 for each vertex i ∈ ( Q T ) given by an interior edge of T .We define the relative Ginzburg algebra G T to be the dg-algebra whose underlying graded algebrais given by the graded path algebra k ˜ Q T and whose differential is determined by the followingaction on the generators. a a ∗ ∂ a W ′ T b i X a ∈ ( Q T ) p i [ a, a ∗ ] p i Note that if the boundary of S is empty, then G T is equivalent to G ( Q ◦ T , W ′ T ). As an example,let S be the 3-gon and T a triangle. The relative Ginzburg algebra G T is then given by the gradedpath algebra of the graded quiver ·· · (1)with differential d mapping each arrow of degree 1 to the composite of the opposite arrows ofdegree 0. The Ginzburg algebra G ( Q ◦ T , W ′ T ) of the triangle T however is zero.Theorem 1 extends to relative Ginzburg algebra in the following way. Theorem 2.
Let T be an ideal triangulation of an oriented marked surface S with dual ribbongraph Γ . The ∞ -category of global sections of the parametrized perverse schober F T satisfies H (Γ , F T ) ≃ D ( G T ) . In [Kel11, Section 7.6], it shown that mutation of quivers with potential induce derivedequivalences between the respective Ginzburg algebras. In [LF09] it is shown that if two idealtriangulations T , T ′ are related by a flip of an edge, the associated quivers with potentials( Q ◦ T , W T ) and ( Q ◦ T ′ , W T ′ ) are related by quiver mutation. In combination these two resultsshow that flips of the ideal triangulation induce derived equivalences of the associated Ginzburgalgebras. We will show that such derived equivalences also exist between the relative Ginzburgalgebras. heorem 3. Let S be an oriented marked surface with two ideal triangulation T , T ′ related bya flip of an edge e of T . Then there exists an equivalence of ∞ -categories µ e : D ( G T ) ≃ D ( G ′ T ) . The proof of Theorem 3 consists of a simple argument using an intrinsic feature of the theoryof perverse schobers, see Section 6.4.We thank Bernhard Keller for informing us about an alternative approach to Theorem 3.A result of Yilin Wu [Wu] extends the argument from [Kel12, Section 7.6] to relative Ginzburgalgebras, showing that mutations of ice quivers with potential induce derived equivalences be-tween the associated relative Ginzburg algebras. The result assumes that the ice quiver whichis mutated at a vertex v has no 2-cycles incident to v . Most cases of Theorem 3 would thusfollow if one extends the result of [LF09] relating flips of the ideal triangulation and mutationsof quivers with potentials to ice quivers. A discussion of mutations of ice quivers with potentialcan be found in [Pre20]. Perverse schobers are a conjectured categorification of the notion of perverse sheaf [KS14]. Anapproach to the categorification of a perverse sheaf on a disk was suggested in [KS14]. The datumof a perverse sheaf on a disk with a single singularity in the center is equivalent to the datumof a certain quiver diagram; the proposed ’ad-hoc’ categorification of the quiver description is aspherical adjunction. In this paper, we extend this ad-hoc categorification to perverse schoberson marked surfaces. We combinatorially describe perverse schober using ribbon graphs. Such aribbon graph arises as the dual to an ideal triangulation of the marked surface. Given a ribbongraph Γ we define a poset Exit(Γ) with• objects the vertices and edges of Γ,• morphism of the form v → e with v a vertex and e an incident edge.For each n -valent vertex v of Γ there exists a subposet Exit(Γ) v/ ⊂ Exit(Γ) consisting of thevertex v and the n incident edges. We define a perverse schober F parametrized by Γ to be afunctor F : Exit(Γ) → St into the ∞ -category of stable ∞ -categories such that the restrictionto Exit(Γ) v/ is for every vertex v equivalent to a particular diagram obtained from a sphericaladjunction. The exact definition is based on the categorified Dold-Kan correspondence of [Dyc17]and categorifies the ’fractional spin’ description of perverse sheaves on a disc of [KS16a]. Thedefinition of parametrized perverse schober captures the idea that a perverse schober on a surfaceis a collection of suitably glued together spherical adjunctions. The ∞ -category of global sections H (Γ , F ) of a parametrized perverse schober F is defined as the limit over F in St and is undermild technical assumptions equivalent to a suitable colimit of the left adjoint diagram. The ∞ -category of global sections H Γ ◦ (Γ , F ) with support on the interior Γ ◦ of Γ is defined as acertain full subcategory of H (Γ , F ), see Definition 4.18Given an ideal triangulation T without self-folded triangles of such a surface S , Smith [Smi15]defines a Calabi-Yau 3-fold Y with a smooth map π : Y → S . The relation to Ginzburg algebrasis as follows, see loc. cit .• The derived category of finite modules over G ( Q ◦ T , W ′ T ) arises as a full subcategory of thederived Fukaya category Fuk( Y ) of Y .• The derived category of finite modules over G ( Q ◦ T , W T ) is a full subcategory of the derivedFukaya category Fuk( Y, b ) of Y with a twisting background class b ∈ H ( Y, Z ).The geometry of the fibers of π can be described as follows.• The generic fiber of π is diffeomorphic to T ∗ S . For each ideal triangle of T , there exists a point with fiber given by the 2-dimensional A -singularity.• The fiber of an interior vertex of T is given by C ∐ C .The twist by the background class b ∈ H ( Y, Z ) changes signs in the signed count of pseudo-holomorphic curves passing through the fibers of the interior vertices. Without the backgroundclass the signed count of such pseudo-holomorphic curves always vanishes so that the derivedFukaya category of Y ◦ := π − (Σ) is equivalent to the derived Fukaya category of Y , whereΣ = S \ ( M ∩ S ◦ ) denotes the open surface. The change in the A ∞ -structure of the derivedFukaya category of Y induced by the background class b accounts exactly for the differencebetween the potentials W ′ T and W T .The restriction π | Y ◦ : Y ◦ → Σ of π is a Lefschetz fibration. As explained in the introduction,we expect the parametrized perverse schober F T of Theorem 1 to be closely related to a Fukayacategory of Y ◦ . In the case of the unpunctured n -gon, where Y ◦ = Y is the 3-dimensional A n − -singularity and Q ◦ T the A n − -quiver, it is shown in [LU20] that W ( Y ◦ ) ≃ D ( G ( Q ◦ T , W ′ T )) perf ,meaning that H Γ ◦ (Γ , F T ) is equivalent to the Ind-completion of the wrapped Fukaya categoryof Y ◦ .We describe in Section 1.3 how the geometry of the Lefschetz fibration manifests itself inthe definition of F T . We expect that the that the twisting by the background class b can bedescribed as a deformation of the (wrapped) Fukaya category. It would be interesting to studythe relation between such a deformation and the description in terms of parametrized perverseschobers. We now describe the construction of the perverse schober F T appearing in Theorem 1 andTheorem 2. We assume for simplicity that all ideal triangles of T are not self-folded. The ribbongraph Γ dual to T parametrizing F T consists of a vertex for each interior ideal triangle and anedge for each edges of T . Boundary edges of T correspond to external edges of the ribbon graph.Parametrized perverse schobers can, as do sheaves, be glued. To define F T , it thus suffices todefine F T locally at each singular vertex of Γ. The local datum at each singular vertex is aspherical adjunction. At each vertex, we choose the spherical adjunction f ∗ : D ( k ) ←→ Fun( S , D ( k )) : f ∗ where Fun( S , D ( k )) is the ∞ -category of local systems on the 2-sphere with values in D ( k ) and f ∗ is the pullback functor along S → ∗ . This adjunction was shown in [Chr20] to be spherical.The ∞ -category Fun( S , D ( k )) is equivalent to the derived ∞ -category of the polynomial algebra k [ t ] with generator t in degree 1, see Proposition 5.5, which is equivalent to the Ind-completionof the wrapped Fukaya of the cotangent bundle T ∗ S , see [Abo11]. The spherical adjunction f ∗ ⊣ f ∗ thus reflects the geometry of the Lefschetz fibration π | Y ◦ whose singular fibers are eachgiven by the A -singularity (which is topologically contractible).The parametrized perverse schober F T in total corresponds to the datum of a diagram F T : Exit(Γ) → Stin the ∞ -category St of stable ∞ -categories indexed by the poset Exit(Γ), see Section 1.2. Thecomputations in Section 5 show that the parametrized perverse schober F T assigns• to each vertex of Γ T a stable ∞ -category equivalent to the derived ∞ -category of therelative Ginzburg algebra of the 3-gon, depicted in diagram (1). This uses that eachvertex of Γ T is trivalent. to each edge of Γ T a stable ∞ -category equivalent to the derived ∞ -category of the poly-nomial algebra k [ t ] with generator t in degree 1. Note that k [ t ] is equivalent to the2-Calabi-Yau completion of k in the sense of [Kel11], i.e. a 2-dimensional Ginzburg alge-bra.The equivalence H (Γ , F T ) ≃ D ( G T ) of Theorem 2 thus expresses that the derived ∞ -categoryof the relative Ginzburg algebra G T is glued from relative Ginzburg algebras of 3-gons along2-dimensional Ginzburg algebras. Acknowledgements
I thank my supervisor Tobias Dyckerhoff for proposing this topic and for his support and en-couragement. I further thank Dylan Allegretti and Bernhard Keller for helpful comments.
This paper is formulated using the language of stable ∞ -categories. It would in principle bepossible to formulate most results in the framework of dg-categories. Our reason to use stable ∞ -categories is to gain access to the powerful framework developed in [Lur09, Lur17]. As aside effect, we also profit in Section 7.2 from the added generality of stable ∞ -categories overdg-categories. The essential computations in the gluing construction for the Ginzburg algebrasare however performed using the category of dg-categories with its Morita model structure. Thereader without a strong background in ∞ -categories and with a particular interest in the gluingconstruction may thus wish to directly skip to the examples in Section 6.2 and to Section 6.1.The goal of this section is to review background material on the relation between on the onehand ring spectra, stable ∞ -categories and their colimits and on the other hand dg-algebras,dg-categories and their homotopy colimits. All material appearing in this section for which wecould not find references in the literature is well known to experts. In Sections 2.1 and 2.2we discuss some generalities on limits and colimits in ∞ -categories of ∞ -categories and on ∞ -categories of modules associated to ring spectra. In Sections 2.3 to 2.5 we relate dg-categorieswith ∞ -categories. In Section 2.6 we discuss semiorthogonal decompositions.For a first introduction to the theory of stable ∞ -categories in a related context, we referto [Chr20, Section 1.1]. For an extensive treatment of the theory of ∞ -categories and stable ∞ -categories we refer to [Lur09] and [Lur17], respectively. ∞ -categories of ∞ -categories We begin by introducing the following ∞ -categories of ∞ -categories. Definition 2.1.
We denote1. by Cat ∞ the ∞ -category of ∞ -categories.2. by St ⊂ Cat ∞ the subcategory spanned by stable ∞ -categories and exact functors.3. by St idem ⊂ St the full subcategory spanned by idempotent complete stable ∞ -categories.An ∞ -category is called presentable if it is equivalent to the Ind-completion of a small ∞ -category and admits all colimits , see [Lur09, Section 5.5]. We further denote4. by P r L ⊂ Cat ∞ the subcategory spanned by presentable ∞ -categories and colimit pre-serving functors.5. by P r R ⊂ Cat ∞ the subcategory spanned by presentable ∞ -categories and accessible andlimit preserving functors. We always assume all limits and colimits to be small in the sense of [Lur09]. . by P r L St ⊂ P r L and P r R St ⊂ P r R the full subcategories spanned by stable ∞ -categories.We are further interested in R -linear ∞ -categories, where R is an E ∞ -ring spectrum, i.e. acommutative algebra object in the symmetric monoidal ∞ -category Sp of spectra. The ∞ -category P r L also admits the structure of a symmetric monoidal ∞ -category, see [Lur17, Section4.8.1]. Given an E ∞ -ring spectrum R , the ∞ -category LMod R ∈ P r L of left module-spectraover R is an algebra object of P r L . Definition 2.2.
7. Let R be an E ∞ -ring spectrum. The ∞ -category of LinCat R = LMod LMod R ( P r L ) of leftmodules in P r L over LMod R , is called the ∞ -category of R -linear ∞ -categories. Remark 2.3.
Though not directly contained in the definition, it can be shown that any R -linear ∞ -category is automatically stable, see [Lur18, D.1.5] for a discussion. Remark 2.4.
A left-tensoring of an ∞ -category M over a monoidal ∞ -category C ⊗ is a co-Cartesian fibration of ∞ -operads O ⊗ → LM ⊗ over the left-module ∞ -operad LM ⊗ , such thatthere are equivalences of fibers O ⊗h m i ≃ M and O ⊗h a i ≃ C ⊗ . We refer to [Lur17, Section 4.2.1]for more details. Objects of LinCat R can be identified with stable and presentable ∞ -category C with the datum of a left-tensoring over the symmetric monoidal ∞ -category LMod R , suchthat the tensor product - ⊗ R - : LMod R × C → C preserves colimits separately in each variable,see [Lur18, Appendix D]. Let M , M be R -linear ∞ -categories as witnessed by the coCartesianfibrations O ⊗ , O ⊗ → LM ⊗ . An R -linear functor M → M thus corresponds to a morphism of ∞ -operads O ⊗ → O ⊗ over LM ⊗ .We now recall in order of appearance results oni) how to compute limits in Cat ∞ ,ii) how to compute limits and colimits in P r L , P r L St and P r R , P r R St ,iii) how to compute limits and colimits in LinCat R andiv) how to compute limits and colimits in St idem .i) There is a general formula for limits in Cat ∞ . Let D : Z → Set ∆ be a diagram takingvalues in ∞ -categories. Consider the coCartesian fibration p : X → Z classified by D . The limit ∞ -category lim D is equivalent to the ∞ -category of coCartesian sections of p , see [Lur09,3.3.3.2]. If Z is the nerve of a 1-category, the above model for computing limits in Cat ∞ canbe described more explicitly. We can use the relative nerve construction, see [Lur09, 3.2.5.2],for the coCartesian fibration classified by D , which is very explicitly defined. We denote thismodel for the coCartesian fibration by p : Γ( D ) → K and call it the covariant Grothendieckconstruction. A more detailed introduction to the relative nerve construction can be found inSection 1.2 of [Chr20].ii) One of the nice features of presentable ∞ -categories is that there is an ∞ -categoricaladjoint functor theorem, which states that a functor between presentable ∞ -categories admitsa right adjoint if and only if it preserves all colimits and admits a left adjoint if and only if it isaccessible and preserves all limits. There thus exists an adjoint equivalence of ∞ -categoriesradj : P r L ≃ (cid:0) P r R (cid:1) op : ladj op , with the functors radj , ladj acting as the identity on objects. The functor radj maps a colimitpreserving functor to its right adjoint and the functor ladj maps an accessible and limit preserv-ing functor to its left adjoint. The adjoint equivalence radj ⊣ ladj also restricts to an adjoint We call a section s : Z → X of a coCartesian fibration p : X → Z coCartesian if for all edges e ∈ Z , the edge s ( e ) ∈ X is s -coCartesian. The functor between presentable ∞ -categories being accessible reduces to the condition of preserving filteredcolimits. quivalence between P r L St and ( P r R St ) op . The equivalences radj , ladj preserve all limits and col-imits, so that we can exchange the computations of limits and colimits of diagrams of (stable)presentable ∞ -categories. For the computation of limits, we can use i) and the fact that thatthe inclusions P L St ⊂ P r L ⊂ Cat ∞ and P r R St ⊂ P r R ⊂ Cat ∞ preserve all limits.iii) The computation of limits and colimits of R -linear ∞ -categories reduces to the compu-tation of limits and colimits in P r L , because the forgetful functor LMod LMod R ( P r L ) → P r L preserves all limits and colimits, see [Lur17, 4.2.3.1,4.2.3.5].iv) The inclusion functor St idem ⊂ Cat ∞ preserves all limits. The computation of colimitsof idempotent stable ∞ -categories can be related to the computation of colimits of presentablestable ∞ -categories via the colimit preserving Ind-completion functor Ind : St idem → P r L St andthe colimit preserving functor (-) c : P r L St → St idem assigning to a stable and presentable ∞ -category the full subcategory of compact objects. The composite functor (-) c ◦ Ind is equivalentto the identity functor.
Consider the symmetric monoidal ∞ -category Sp of spectra. Sp is a stable and presentable ∞ -category. An E -ring spectrum is an object of Alg(Sp), the ∞ -category of (coherently as-sociative) algebra objects in Sp. For every such E -ring spectrum R , there is a stable andpresentable ∞ -category RMod R of right R -modules in Sp. If R can be enhanced to a com-mutative algebra object of Sp, i.e. an E ∞ -ring spectrum, then RMod R inherits the structureof a symmetric monoidal ∞ -category. In this case, we can form the ∞ -category Alg(RMod R )of algebra objects in RMod R . Given A ∈ Alg(RMod R ), we can again form the ∞ -categoryRMod A (RMod R ) of right A -modules in RMod R . Alternatively, we can also consider the E -ring spectrum ξ ( A ) ∈ Alg(Sp) underlying A . Formally, we consider the forgetful functorRMod R → Sp, mapping a right R -module to the underlying spectrum. This functor extends toa functor ξ : Alg(RMod R ) → Alg(Sp), which we apply to A . We can then form the ∞ -categoryof right modules RMod ξ ( A ) over ξ ( A ). We will show in Corollary 2.7 that this does not yield afurther ∞ -category, there exists an equivalence of ∞ -categoriesRMod A (RMod R ) ≃ RMod ξ ( A ) . Let D be a stable ∞ -category and consider any object X ∈ D . We can find an E -ringspectrum End( X ) ∈ Alg(Sp), called the endomorphism algebra with the following properties,see [Lur17, 7.1.2.2].• π n End( X ) ≃ π Map D ( X [ n ] , X ) for all n ∈ Z .• The induced ring structure of π ∗ End( X ) is determined by the composition of endomor-phisms in the homotopy category Ho( D ).The algebra object End( X ) is an endomorphism object of X in the sense of [Lur17, Section 4.7.1].The existence of End( X ) expresses the enrichment of the stable ∞ -category D in spectra.Assume that the stable ∞ -category D is also presentable. An object X ∈ D is called acompact generator if• X is compact, i.e. Map D ( X, -) commutes with filtered colimits and• an object Y ∈ D is zero if and only if Map D ( X, Y [ i ]) ≃ ∗ for all i ∈ Z .The importance of the notion of compact generator is that if X is a compact generator, thereexists an equivalence of ∞ -categories D ≃ RMod
End( X ) , see [Lur17, 7.1.2.1].We now restrict to R -linear ∞ -categories where R is an E ∞ -ring spectrum. The mostimportant case will be where R = k is a commutative ring. Suppose that D is an R -linear ∞ -category and X ∈ D a compact generator. Lemma 2.5 shows we can lift End( X ) along theforgetful functor ξ : Alg(RMod R ) → Alg(Sp) to an algebra object in RMod R . emma 2.5. Let R be an E ∞ -ring spectrum. Let C be a stable and presentable R -linear ∞ -category with a compact generator X . Then there exists an algebra object End R ( X ) ∈ Alg(RMod R ) and an equivalence of R -linear ∞ -categories C ≃ RMod
End R ( X ) (RMod R ) . (2) The algebra object
End R ( X ) is mapped under the functor ξ : Alg(RMod R ) → Alg(Sp) to theendomorphism algebra
End( X ) ∈ Alg(Sp) .Proof.
The left tensoring of C over R determined an R -linear functor - ⊗ R X : RMod R → C .By the adjoint functor theorem, the functor admits a right adjoint G . We denote End R ( X ) := G ( X ) ∈ RMod R . The existence of lift of End R ( X ) to Alg(RMod R ) and the existence of theequivalence (2) follow from [Lur17, 4.8.5.8], compare also to the proof of [Lur17, 7.1.2.1]. Theright adjoint of the composite functorSp - ⊗ R −−−→ RMod R - ⊗ R X −−−−→ C maps X to the endomorphism object End( X ). By the universal property of End( X ) and X ∈ C ≃ RMod ξ (End R ( X )) (Sp), there exists a morphism ξ (End R ( X )) → End( X ) in Alg(Sp), whichis an equivalence on underlying spectra and thus an equivalence of E -ring spectra. Remark 2.6.
In the setting of Lemma 2.5, the algebra object End R ( X ) is an endomorphismobject of X in the ∞ -category C considered as left tensored over RMod R . We call End R ( X ) the R -linear endomorphism algebra of X . Corollary 2.7.
Let R be an E ∞ -ring spectrum and A ∈ Alg(RMod R ) . Then there exists anequivalence of ∞ -categories RMod A (RMod R ) ≃ RMod ξ ( A ) , where ξ : Alg(RMod R ) → Alg( Sp ) denotes the forgetful functor.Proof. The ∞ -category RMod A (RMod R ) is presentable by [Lur17, 4.2.3.7], stable by [Lur17,7.1.1.4] and left-tensored over RMod R by [Lur17, Section 4.3.2]. Consider the monadic ad-junction - ⊗ A : RMod R ↔ RMod A (RMod R ) : G . The adjunction and that G is conservativeand accessible imply that A is a compact generator. The R -linear endomorphism algebra of A ∈ RMod A (RMod R ) is given by A ∈ Alg(RMod R ). The statement thus follows from thesecond part of Lemma 2.5 and [Lur17, 7.1.2.1].Let R be an E ∞ -ring spectrum. We end this section with a brief discussion of the relationbetween colimits of algebra objects in RMod R and the colimits of the corresponding ∞ -categoriesof right modules in LinCat R . There is a functor θ : Alg(RMod R ) → LinCat R that assigns to analgebra object A ∈ Alg(RMod R ) the ∞ -category RMod A (RMod R ), see [Lur17, section 4.8.3].The functor θ assigns to an edge φ : A → B in Alg(RMod R ) the relative tensor product θ ( φ ) = - ⊗ A B : RMod A (RMod R ) −→ RMod B (RMod R )using the right A -module structure on B provided by φ . For all φ : A → B , the functor θ ( φ )admits a right adjoint, given by the pullback functor φ ∗ : RMod B (RMod R ) → RMod A (RMod R )along φ , see [Lur17, 4.6.2.17]. The functor θ preserves colimits indexed by contractible simpli-cial sets (i.e. simplicial sets whose geometric realization is a contractible space), most notablypushouts. .3 Differential graded categories and their modules Let k be a commutative ring. A k -linear dg-category is a 1-category enriched in the 1-categoryCh( k ) of chain complexes of k -modules. Given a dg-category C and two objects x, y ∈ C , wewrite Hom C ( x, y ) or Hom( x, y ) for the Hom complex. We consider dg-algebras as dg-categorieswith a single object. Definition 2.8.
Let A and B be k -linear dg-algebras.• A left A -module M is a graded left module over the graded algebra underlying A equippedwith a differential d M such that d M ( a.m ) = d A ( a ) .m + ( − deg( a ) a.d M ( m )for all a ∈ A and m ∈ M .• A right A -module M is a graded right module over the graded algebra underlying A equipped with a differential d M such that d M ( m.a ) = d M ( m ) .a + ( − deg( m ) m.d A ( a )for all a ∈ A and m ∈ M . We also refer to right A -modules simply as A -modules.• An A - B -bimodule M is a graded bimodule over the graded algebras underlying A and B equipped endowed with a differential d M , which exhibits M as a left A -module and a right B -module. If A = B , we call M an A -bimodule. Remark 2.9.
Let M be an A - B -bimodule with differential d M . The shifted A - B -bimodule M [1] can be described as follows.• The differential is − d M .• The left action . [1] of a ∈ A on m ∈ M [1] is given by a. [1] m = ( − deg( a ) a.m , where a.m denotes the left action of a ∈ A on m ∈ M .• The right action . [1] of b ∈ B on m ∈ M [1] is given by m. [1] b = m.b , where m.b denotesthe right action of b ∈ B on m ∈ M .We can identify left A -modules with dg-functors A → Ch( k ) and right A -modules withdg-functors A op → Ch( k ). The following definition is thus consistent with Definition 2.8. Definition 2.10.
Let C be a dg-category. We call a dg-functor C op → Ch( k ) a right C -module.We denote by dgMod( C ) the dg-category of right C -modules. Remark 2.11.
Given any dg-category C , the dg-category dgMod( C ) is pretriangulated, withdistinguished triangles of the form x a −→ y → cone( a ).Given a dg-category C and an object x ∈ C , we denote by End dg ( x ) the endomorphism dg-algebra with underlying chain complex given by Hom C ( x, x ) and algebra structure determinedby the composition of morphisms in C . Lemma 2.12.
Let C be a dg-category with finitely many objects x , . . . , x n . Then there existsan equivalence of dg-categories dgMod( C ) ≃ dgMod(End dg ( L ni =1 x i ) , where End dg ( L ni =1 x i ) isthe endomorphism dg-algebra of L ni =1 x i in dgMod( C ) .Proof. This follows directly from spelling out the datum of a right module over C and overEnd dg ( L ni =1 x i )s. .4 A model for the derived ∞ -category of a dg-algebra Let A be a k -linear dg-algebra. The 1-category dgMod( A ) underlying the dg-category dgMod( A )is the 1-category with the same objects and with mapping sets given by the 0-cycles. This 1-category admits the projective model structure, where the weak equivalences are the givenby quasi-isomorphisms and the fibrations are given by degree-wise surjections. All objects ofdgMod( A ) are fibrant. A description of the cofibrant objects in dgMod( A ) can be found ex-ample in [BMR14], where they are called q -semi-projective objects. A right A -module M iscofibrant if and only if• the ungraded module L i ∈ Z M i is a projective right module over the ungraded algebra L i ∈ Z A i and• for all acyclic right A -modules N , the Hom complex Hom A ( M, N ) is acyclic.If A = k is a commutative ring, the cofibrant objects are the complexes of projective k -modules.We denote by dgMod( A ) ◦ ⊂ dgMod( A ) the full dg-subcategory spanned by fibrant-cofibrantobjects. We call the dg-nerve D ( A ) := N dg (dgMod( A ) ◦ ) the (unbounded) derived ∞ -categoryof A .Before we can further discuss the properties of D ( A ), we need to briefly discuss localizationsof ∞ -categories. Definition 2.13.
A functor f : C → C ′ between ∞ -categories is a reflective localization if f hasa fully faithful right adjoint.In [Lur09], localizations in the sense of Definition 2.13 are simply called localizations. Weare however interested in a more general class of localizations, which can be characterized bythe following universal property. Definition 2.14.
Let C be an ∞ -category and let W be a collection of morphisms in C . Wecall an ∞ -category C ′ the ∞ -categorical localization of C at W if there exists exists a functor f : C → C ′ such that for every ∞ -category D composition with f induces a fully faithful functor χ : Fun( C ′ , D ) → Fun( C , D ) , such that the essential image of χ consists of those functors F : C → D for which F ( α ) is anequivalence in D for all α ∈ W . In that case, we also write C ′ = C [ W − ].It is shown in [Lur09, 5.2.7.12], that reflective localization are localizations in the sense ofDefinition 2.14. If the collection of morphisms W is closed under homotopy and compositionand contains all equivalences in C we can regard C [ W − ] as a fibrant replacement of ( C , W ) inthe model category of marked simplicial sets, see also the discussion in the beginning of [Lur17,Section 4.1.7].Our first goal in this section is to prove the following analogue of [Lur17, 1.3.5.15] for de-rived ∞ -categories of dg-algebras, relating the derived ∞ -category of A with the ∞ -categoricallocalization of dgMod( A ) at the collection of quasi-isomorphisms. Proposition 2.15.
Let A be a dg-algebra and let W denote the collection of quasi-isomorphisms.There exists an equivalence of ∞ -categories D ( A ) ≃ N (dgMod( A ) )[ W − ] . Given a model category C , the ∞ -categorical localization of N ( C ) at the collection of weakequivalences is called the ∞ -category underlying C . We refer to [Hin16] for general background.Proposition 2.15 thus shows that the derived ∞ -category of A is the ∞ -category underlying themodel category dgMod( A ) .For the proof of Proposition 2.15 we need the following two lemmas. emma 2.16. Let A be a dg-algebra. The inclusion functor N (dgMod( A ) ) → N dg (dgMod( A )) induces an equivalence of ∞ -categories N (dgMod( A ) )[ H − ] → N dg (dgMod( A )) , where H is the collection of chain homotopy equivalences.Proof. The proof of [Lur17, 1.3.4.5] applies verbatim.
Lemma 2.17.
Let A be a dg-algebra. There exists an equivalence of ∞ -categories N dg (dgMod( A ) ◦ ) ≃ N dg (dgMod( A ))[ W − ] . Proof.
We adapt the proofs of [Lur17, 1.3.4.6, 1.3.5.12]. We show that the inclusion functor i :N dg (dgMod( A ) ◦ ) op → N dg (dgMod( A )) op admits a left adjoint which exhibits N dg (dgMod( A ) ◦ ) op as a reflective localization at the collection of quasi-isomorphisms. The universal property of thelocalization, see [Lur17, 5.2.7.12], is irrespective of taking opposite categories. We thus concludethat N dg (dgMod( A ) ◦ ) is equivalent as an ∞ -category to the localization of N dg (dgMod( A )) atthe collection of quasi-isomorphisms.We need to show that the inclusion functor i admits a left adjoint G : N dg (dgMod( A )) op → N dg (dgMod( A ) ◦ ) op such that any edge e : M → N in N dg (dgMod( A )) op is a quasi-isomorphismif and only if G ( e ) is an equivalence. Consider a trivial fibration f : Q ′ → Q in dgMod( A )given a cofibrant replacement and any P ∈ dgMod( A ) ◦ . [Lur09, 5.2.7.8] shows the existence of G , provided that, for f considered as a morphism in N dg (dgMod( A )) and P considered as anobject of N dg (dgMod( A ) ◦ ), the composition with f induces an isomorphism of spacesMap N dg (dgMod( A ) ◦ ) ( P, Q ′ ) → Map N dg (dgMod( A )) ( P, Q ) . We deduce this from the assertion that composition with f induces a quasi-isomorphism α : Hom dgMod( A ) ( P, Q ′ ) → Hom dgMod( A ) ( P, Q ) . (3)The surjectivity of α follows from the lifting property of the cofibration 0 → P with respectto trivial fibrations. The kernel of α is given by Hom dgMod( A ) ( P, ker( f )). Using that f is aquasi-isomorphism, we deduce that ker( f ) is acyclic. The contractibility of the kernel of α thusfollows from property of P being cofibrant. We can thus deduce the existence of G . We notethat G is pointwise given by choosing a cofibrant replacement. Consider an edge e : M → N in N dg (dgMod( A )) op . If e is a quasi-isomorphism, it follows from Whitehead’s theorem formodel categories that G ( e ) is an equivalence. If G ( e ) is an equivalence, we have the followingcommutative diagram in N dg (dgMod( A )). G ( M ) G ( N ) M N G ( e ) e The vertical edges and the upper horizontal edge are quasi-isomorphisms. It follows that e isalso a quasi-isomorphism. Proof of Proposition 2.15.
By Lemmas 2.16 and 2.17, there exists an equivalence of ∞ -categories (cid:0) N (dgMod( A ))[ H − ] (cid:1) [ W − ] ≃ D ( A ) . Using that H ⊂ W , the statement follows. et k be a commutative ring. The symmetric monoidal structure of the 1-category Ch( k )can be used to also endow the ∞ -category D ( k ) with a symmetric monoidal structure. As shownin [Lur17, 7.1.4.6] there exists an equivalence of ∞ -categoriesN(Alg(Ch ⊗ ( k )))[ W − ] ≃ Alg( D ( k )) . (4)The left side of (4) is the ∞ -categorical localization of the nerve of the 1-category of dg-algebrasat the collection of quasi-isomorphisms. The right side of (4) is the ∞ -category of algebraobjects in D ( k ). The equivalence (4) expresses that every dg-algebra can be considered as analgebra object in D ( k ) and that every algebra object in D ( k ) can be obtained this way (meaningit can be rectified). Unless stated otherwise, we will omit the identification (4) and considerdg-algebras as algebra objects in the symmetric monoidal ∞ -category D ( k ).We can consider k also as an E ∞ -ring spectrum. The ∞ -category RMod k of right modulesover k thus inherits a symmetric monoidal structure. The ∞ -categories D ( k ) and RMod k areequivalent as symmetric monoidal ∞ -categories, see [Lur17, 7.1.2.13].Let A be a k -linear dg-algebra and X a cofibrant A -module. Consider the Quillen adjunction- ⊗ dg k X : dgMod( k ) ↔ dgMod( A ) : Hom A ( X, -) , (5)between the tensor functor on the level of chain complexes and the internal Hom functorcomposed with the forget functor dgMod( A ) → dgMod( k ). Given a Quillen-adjunction be-tween model categories, there is an associated adjunctions between the underlying ∞ -categories,see [MG16]. We denote the adjunction of ∞ -categories underlying the Quillen adjunction (5)by - ⊗ dg k X : D ( k ) ↔ D ( A ) : RHom A ( X, -) . (6) Lemma 2.18.
Let A be a k -linear dg-algebra. The ∞ -category D ( A ) admits the structure of a k -linear ∞ -category such that for any X ∈ D ( A ) the functor - ⊗ dgk X is k -linear.Proof. The ∞ -category D ( A ) is stable and presentable by [Lur17, 1.3.5.9, 1.3.5.21]. We nowshow D ( A ) is left tensored over D ( k ). Note that dgMod( k ) ≃ Ch( k ) is a symmetric monoidalmodel category with respect to the tensor product, which we denote in the following by ⊗ , see[Lur17, 7.1.2.11]. We further denote the Quillen bifunctor dgMod( k ) × dgMod( A ) → dgMod( A )given by the relative tensor product by - ⊗ dg k -. Recall that LM ⊗ denotes the left-module ∞ -operad, see [Lur17, 4.2.1.7]. We define a 1-category O ⊗ A as follows.• An object of O ⊗ A consists of an object ( a, . . . , a | {z } i -many , m, . . . , m | {z } j -many ) ∈ LM ⊗ and objects( x , . . . , x i ) ∈ (dgMod( k ) ◦ ) × i , ( m , . . . , m i ) ∈ (dgMod( A ) ◦ ) × j . • For n = 1 ,
2, consider the object X n of O ⊗ A given by l n = ( a, . . . , a | {z } i n -many , m, . . . , m | {z } j n -many ) ∈ LM ⊗ and( x n , . . . , x ni n ) ∈ (dgMod( k ) ◦ ) × i n , ( m n , . . . , m nj n ) ∈ (dgMod( A ) ◦ ) × j n . A morphism X → X consists of a morphism α : l → l in LM ⊗ , which we also consideras a morphism of sets ˜ α : { , . . . , i + j } → { , . . . , i + j } , morphisms O e ∈ ˜ α − ( i ) a e → a i n dgMod( k ) ◦ for 1 ≤ i ≤ i and morphisms (cid:18) O e ∈ ˜ α − ( j ) \ max(˜ α − ( j )) a e (cid:19) ⊗ k m α − ( j )) − i → m j − i in dgMod( A ) ◦ for i + 1 ≤ j ≤ i + j .The forgetful functor N ( O ⊗ A ) → LM ⊗ is a coCartesian fibration of ∞ -operads, exhibiting N ((dgMod( A ) ◦ ) ) as left-tensored over the symmetric monoidal ∞ -category N ((dgMod( k ) ◦ ) ).By the discussion following [Lur17, 4.1.7.3] and using that - ⊗ dg k - preserves weak equivalencesin both entries, it follows that the left-tensoring passes to the ∞ -categorical localizations atthe chain homotopy equivalences, meaning that we obtain that D ( A ) is left-tensored over D ( k ).The action of D ( k ) on D ( A ) preserving colimits in both variables, as follows from the monoidalproduct - ⊗ k - being a Quillen-bifunctor. To see that - ⊗ dg k X is a k -linear functor, we need todescribe an extension of - ⊗ dg k X to a map α : N ( O ⊗ k ) → N ( O ⊗ A ) of ∞ -operads over LM ⊗ . Weleave the details of the description of a functor of 1-categories α ′ : O ⊗ k → O ⊗ A whose nerve N ( α ′ )defines the desired functor α to the reader. Proposition 2.19.
Let A be a k -linear dg-algebra. Using the symmetric monoidal equivalence D ( k ) ≃ RMod k , we can consider RMod A as left-tensored over D ( k ) . There exists an equivalence D ( A ) ≃ RMod A (7) of ∞ -categories left-tensored over D ( k ) .Proof. Consider the adjunction of ∞ -categories ⊗ dg k A : D ( k ) ↔ D ( A ) : RHom A ( A, -) underlyingthe Quillen adjunction ⊗ dg k A : dgMod( k ) → dgMod( A ) : Hom A ( A, -). Using the adjunction itcan be directly checked that A is a compact generator of D ( A ). It follows from [Lur17, 4.8.5.8]that there exists an equivalence D ( A ) ≃ RMod
End k ( A ) (8)of ∞ -categories left-tensored over D ( k ), where End k ( A ) ∈ Alg( D ( k )) is the k -linear endomor-phism algebra of A , see Remark 2.6. We note that the underlying chain complex satisfiesEnd k ( A ) = RHom A ( A, A ) ≃ A . By the universal property of End k ( A ), there exists a morphismof dg-algebras χ : A → End k ( A ), the underlying morphism of chain complexes of which isinduced by the actions A ⊗ k A → A and A ⊗ k End k ( A ) → A . The latter is induced by thecounit of the adjunction ⊗ dg k A ⊣ RHom A ( A, -) and thus equivalent to the former. It followsthat χ induces an the equivalence End k ( A ) = RHom A ( A, A ) ≃ A on underlying chain com-plexes and is hence an equivalence of dg-algebras. In total we obtain, that there also exists anequivalence of k -linear ∞ -categories RMod End k ( A ) ≃ RMod A , which combined with (8) showsthe statement.Let A, B ∈ Alg( D ( k )) be dg-algebras and F : RMod A → RMod B a k -linear functor. Clearly F ( A ) ∈ RMod B carries the structure of a right B -module. Let m : A ⊗ k A → A be themultiplication map of A . Using the k -linearity of F , we find an action map A ⊗ k F ( A ) ≃ F ( A ⊗ k A ) F ( m ) −−−→ F ( A ) , which is part of the datum of a left A -module structure on F ( A ). It turns out that both modulestructures are compatible, so that we can endow F ( A ) with the structure of an A - B -bimodule.In total, we obtain a functor φ : Lin k (RMod A , RMod B ) → BMod
A B ( D ( k )) . s is shown in [Lur17, Section 4.8.4], the functor φ is an equivalence of ∞ -categories. Givena bimodule M ∈ BMod
A B ( D ( k )), we denote by - ⊗ A M a choice of k -linear functor such that φ (- ⊗ A M ) ≃ M . Proposition 2.20.
Let
A, B be dg-algebras and M ∈ BMod
A B ( D ( k )) ≃ D ( A op ⊗ k B ) andconsider the functor of ∞ -categories - ⊗ dg A M underlying the right Quillen functor - ⊗ dg A M :dgMod( A ) → dgMod( B ) . There exists a commutative diagram in LinCat k as follows. RMod A RMod B D ( A ) D ( B ) (7) ≃ - ⊗ A M (7) ≃ - ⊗ dg A M (9) Remark 2.21.
As justified by Proposition 2.20, we will not distinguish in notation betweenthe functors - ⊗ A M and - ⊗ dg A M in the remainder of the text. Proof of Proposition 2.20.
The k -linear functor χ : RMod A ≃ D ( A ) - ⊗ dg A M −−−−→ D ( B ) ≃ RMod B is of the form - ⊗ A N for N ∈ BMod
A B ( D ( k )), see [Lur17, Section 4.8.4]. We note that N canbe rectified to a strict dg-bimodule and is thus determined by its right B -module structure andits left A -module structure. The right B -module structures of N and M are clearly equivalent.In particular, there exists an equivalence N ≃ M of underlying chain complexes. The left actionof A on N is determined by A ⊗ k N ≃ χ ( A ⊗ k A ) χ ( m ) −−−→ χ ( A ) ≃ N, where m denotes themultiplication of A and is thus equivalent to the given left action of A on the A - B -bimodule M .This shows that N ≃ M as bimodules. Proposition 2.22.
Let A be a k -linear dg-algebra and X ∈ dgMod( A ) a cofibrant A -module.The k -linear endomorphism algebra End k ( X ) ∈ Alg( D ( k )) of X is quasi-isomorphic to theendomorphism dg-algebra End dg ( X ) of X .Proof. Proposition 2.20 shows that the functor F : D ( k ) ≃ RMod k - ⊗ k X −−−−→ RMod A (RMod k ) ≃ D ( A )is equivalent to - ⊗ dg k X . The right adjoint G of F is given by RHom A ( X, -). It follows thatRHom A ( X, X ) = G ( X ) = End k ( X ) in D ( k ), see also the definition of End k ( X ) in the proof ofLemma 2.5. Using that RHom A ( X, X ) = Hom A ( X, X ) and the explicit Hom A ( X, X )-modulestructure on X , it follows from the universal property of the endomorphism object that thereexists a morphism of dg-algebras α : RHom A ( X, X ) → End k ( X ), which restricts to the equiva-lence on underlying chain complexes and is hence an equivalence of dg-algebras. Let k be a commutative ring. We denote by dgCat k the category of k -linear dg-categories. Givena dg-category C , the dg-category dgMod( C ) admits a model structure called the projectivemodel structure. We have already encountered this model structure in Section 2.4 in the casewhere C is a dg-algebra. We define C perf as the full dg-subcategory of dgMod( C ) spannedby fibrant-cofibrant objects x which are compact in the homotopy category H (dgMod( C )),i.e. Hom( x, -) preserves coproducts. This assignment forms a functor(-) perf : dgCat k → dgCat k . s shown by Tabuada, the category dgCat k admits a model structure, where• the weak equivalences are the quasi-equivalences, that is dg-functors F : A → B suchthat for all a, a ′ ∈ A , the morphism between Hom complexes F ( a, a ′ ) : Hom A ( a, a ′ ) → Hom B ( F ( a ) , F ( a ′ )) is a quasi-isomorphism and such that the induced functor on homotopycategories is an equivalence.• the fibrations are the dg-functors F such that for all a, a ′ ∈ A , the morphism betweenHom complexes F ( a, a ′ ) : Hom A ( a, a ′ ) → Hom B ( F ( a ) , F ( a ′ )) is degreewise surjective andsuch that for any isomorphism b → F ( a ′ ) in the homotopy category of B , there exists alift along F to an isomorphism a → a ′ in the homotopy category of A .The ∞ -category underlying this model category is given by the ∞ -categorical localizationdgCat k [ W − ] of dgCat k (we do not distinguish between categories and their nerves) at the col-lection W of weak equivalences. As also shown by Tabuada, this model structure can be furtherlocalized at the collection M of Morita-equivalences, that is dg-functors F such that ( F ) perf isa quasi-equivalence. The resulting model structure is called the Morita model structure. Thecorresponding localization functor L : dgCat k [ W − ] −→ dgCat k [ M − ]exhibits dgCat k [ M − ] as a reflective localization of dgCat k [ W − ] and thus preserves colimits.Given C ∈ dgCat k [ W − ], its image L ( C ) is quasi-equivalent to C perf . The Morita modelstructure models the ∞ -category of k -linear, stable and idempotent complete ∞ -categories,meaning that there exists an equivalence of ∞ -categoriesdgCat k [ M − ] ≃ Mod N dg ( k perf ) (St idem ) , (10)see [Coh13]. The right side of (10) describes the ∞ -category of modules in the symmetricmonoidal category St idem over the algebra object N dg ( k perf ). The equivalence (10) maps adg-category C to the dg-nerve of the dg-category C perf . Ind-completion provides a furthercolimit preserving functor Ind : Mod D ( k ) perf (St idem ) → LinCat k . In total we obtain the colimitpreserving functor ψ : dgCat k [ W − ] L −−→ dgCat k [ M − ] ≃ Mod D ( k ) perf (St idem ) Ind −−→
LinCat k forget −−−−→ P r L . (11)For later reference we record the following lemma. Lemma 2.23.
Let C be a dg-category with n objects denotes x , . . . , x n . For any ≤ i ≤ n , consider the right C -module M = Hom C (- , x i ) ∈ dgMod( C ) ≃ dgMod(End dg ( L ni =1 x i )) corepresented by x i . Let f : k → C be the unique morphism of dg-categories determined bymapping the unique object ∗ of k to x i . Then the following diagram in LinCat k commutes. D ( k ) D (End dg ( L ni =1 x i )) ψ ( k ) ψ ( C ) ≃ - ⊗ k M ≃ ψ ( f ) Proof.
Using Lemma 2.12 and Proposition 2.20, the statement follows from the observation that f perf ≃ - ⊗ dg M : dgMod( k ) → dgMod( C ). In this section we discuss semiorthogonal decompositions of stable ∞ -categories of length n ≥ n = 2 in [DKSS19]. efinition 2.24. Let V and A be stable ∞ -categories. We call A ⊂ V a stable subcategory ifthe inclusion functor is fully faithful, exact and its image is closed under equivalences. Notation 2.25.
Let V be a stable ∞ -category and A , . . . , A n ⊂ V stable subcategories. Wedenote by h A , . . . , A n i the smallest stable subcategory of V containing A , . . . , A n . Definition 2.26.
Let V be a stable ∞ -category and let A , . . . , A n be stable subcategories of V .Consider the full subcategory D of Fun(∆ n − , V ) spanned by diagrams D : ∆ n − → V satisfyingthe following two conditions.• D ( i ) lies in h A n − i , . . . , A n i for 0 ≤ i ≤ n − D ( i ) → D ( i + 1) in V lies in A n − i − for all 0 ≤ i ≤ n − n -tuple ( A , . . . , A n ) a semiorthogonal decomposition of V of length n if therestriction functor D → V to the vertex n − Definition 2.27.
Let V be a stable ∞ -category and A ⊂ V a stable subcategory. We define• the right orthogonal A ⊥ to be the full subcategory of V spanned by those vertices x ∈ V such that for all a ∈ A the mapping space Map V ( a, x ) is contractible.• the left orthogonal ⊥ A to be the full subcategory of V spanned by those vertices x ∈ V such that for all a ∈ A the mapping space Map V ( x, a ) is contractible.The next lemma shows that semiorthogonal decompositions of length n are simply repeatedsemiorthogonal decompositions of length 2. Lemma 2.28.
Let V be a stable ∞ -category and A i ⊂ V , for ≤ i ≤ n , a stable subcategory. ( A , . . . , A n ) is a semiorthogonal decomposition of V if and only ifi) h A , . . . , A n i = V andii) ( A i , ⊥ A i ) forms a semiorthogonal decomposition of h A i , . . . , A n i for all ≤ i ≤ n − .Proof. For 1 ≤ i ≤ n − ≤ j ≤ n − i −
1, denote by D ij ⊂ Fun(∆ { j,...,n − i } , h A i , . . . , A n i )the full subcategory spanned by diagrams D ij satisfying that• D ij ( l ) lies in h A n − l , . . . , A n i for j ≤ l ≤ n − i ,• the cofiber of D ij ( l ) → D ij ( l + 1) in V lies in A n − l +1 for all j ≤ l ≤ n − i − r i,j : D ij → h A i , . . . , A n i the functor given by the restriction to the vertex n − i .Note that for i < k ≤ n − j −
1, there is a trivial fibration D ij → D in − k × h A k ,..., A n i D kj .Now suppose that ( A , . . . , A n ) is a semiorthogonal decomposition and let D → V be thecorresponding trivial fibration. Condition i ) is immediate. For condition ii ), we need to showthat r i,n − i − is a trivial fibration for all 1 ≤ i ≤ n −
1. Using that pullbacks preserve trivialfibrations, it follows that D ′ = D × V h A i , . . . , A n i → h A i , . . . , A n i (12)is a trivial fibration. We can describe the elements of D ′ as the left Kan extensions along theinclusion ∆ { ,...,n − i } → ∆ n of elements of D i . It thus follows from [Lur09, 4.3.2.15] that therestriction functor D ′ → D i to ∆ { ,...,i } is a trivial fibration. Using that the functor (12) factorsthrough r i, : D i → h A i , . . . , A n i , it follows from the 2/3 property of equivalences that also r i, is a trivial fibration. The following commutative diagram thus shows that r i,n − i − is a trivialfibration. We have thus shown statement ii ). D i D in − i − × h A i +1 ,..., A n i D i +10 D in − i − h A i , . . . , A n i D i +10 h A i +1 , . . . , A n i triv fib r i, y triv fib r i,n − i − r i +1 , (13) e now show that conditions i ) and ii ) imply that ( A , . . . , A n ) is a semiorthogonal decom-position of V . If n = 2, the assertion is obvious. We proceed by induction over n . Assume that( A , . . . , A n ) is a semiorthogonal decomposition of h A , . . . A n i , meaning that r , is a trivialfibration. To show that ( A , . . . , A n ) is a semiorthogonal decomposition of V = h A , . . . , A n i ,we need to show that r , is also a trivial fibration. Condition ii ) implies that r ,n − is a trivialfibration. The diagram (13) for i = 1 thus shows that r is also a trivial fibration.As it turns out, the functoriality data involved in the definition of semiorthogonal decompo-sitions of length 2 is redundant. Lemma 2.29.
Let V be a stable ∞ -category and let A , B be stable subcategories of V . The pair ( A , B ) forms a semiorthogonal decomposition of length of V if and only if1. for all a ∈ A and b ∈ B , the mapping space Map V ( b, a ) is contractible and2. for every x ∈ V , there exists a fiber and cofiber sequence b → x → a in V with a ∈ A and b ∈ B .Proof. This follows from [Lur18, 7.2.0.2].A simple source of semiorthogonal decompositions are sequences of functors between stable ∞ -categories. Lemma 2.30.
Let D : ∆ n − → Cat ∞ be a diagram taking values in stable ∞ -categories,corresponding to n composable functors A F −−→ A F −−→ . . . F n − −−−→ A n .
1. The stable ∞ -category { A , . . . , A n } := Fun ∆ n − (∆ n − , Γ( D )) of sections of p : Γ( D ) → ∆ n − admits a semiorthogonal decomposition ( A , . . . , A n ) oflength n .2. Let R be an E ∞ -ring spectrum. If each F i is an R -linear functor between R -linear ∞ -categories, then the ∞ -category { A , . . . , A n } further inherits the structure of an R -linear ∞ -category such that each inclusion functor ι i : A i → { A , . . . , A n } is R -linear.Proof. We begin by showing statement 1. Consider the simplicial set Z = (cid:0) ∆ × ∆ n − (cid:1) ∐ ∆ { } × ∆ { ,...,n − } (cid:0) ∆ × ∆ n − (cid:1) ∐ · · · ∐ ∆ { ,...,n − } × ∆ { } (cid:0) ∆ n − × ∆ (cid:1) . Let D ′ be the full subcategory of Fun( Z, Γ( D )) spanned by diagrams given by right Kan exten-sions along the inclusion ∆ n − × ∆ → Z of a diagram in { A , . . . , A n } . By [Lur09, 4.3.2.15], therestriction functor D ′ → { A , . . . A n } to ∆ × ∆ n − is a trivial fibration. We can characterizethe elements of D ′ up to equivalence with diagrams in Γ( D ) of the form a a a . . . . . . . . .a n . . . a n a n idid id id atisfying that a i ∈ A i . The restriction functor D ′ → { A , . . . , A n } corresponds in the abovedescription pointwise to the restriction to the rightmost column. The ∞ -category D of Defini-tion 2.26 can be identified with the full subcategory of Fun(∆ n − × ∆ n − , Γ( α )) spanned by leftKan extensions along Z → ∆ n − × ∆ n − of diagrams lying in D ′ . It follows that the restrictionfunctor D → D ′ is a trivial fibration and thus that the restriction functor D → { A , . . . , A n } isa trivial fibration. This shows statement 1.We now show statement 2. Consider the diagram of ∞ -operads over LM ⊗ D ⊗ : O ⊗ F ⊗ −−→ O ⊗ F ⊗ −−→ . . . F ⊗ n − −−−→ O ⊗ n exhibiting the functors F i as R -linear. The morphism of ∞ -operadFun ∆ n − (∆ n − , Γ( D ⊗ )) × Fun(∆ n − , LM ⊗ ) LM ⊗ → LM ⊗ exhibits { A , . . . , A n } as left tensored over M := Fun ∆ n − (∆ n − , Γ( D ⊗ )) × Fun(∆ n − , LM ⊗ ) LM ⊗ × LM ⊗ Assoc ⊗ . Let ˜ D ⊗ : ∆ n − → Cat ∞ be the constant diagram with value LMod ⊗ R . We find M to be equivalentas a monoidal ∞ -category toFun ∆ n − (∆ n − , Γ( ˜ D ⊗ )) × Fun(∆ n − , Assoc ⊗ ) Assoc ⊗ . (14)Pulling back along the monoidal functor LMod ⊗ R → M , assigning to x ∈ LMod ⊗ R the constantsection in (14), we obtain a left-tensoring of { A , . . . , A n } over LMod R . To show that the left-tensoring provides the structure of an R -linear ∞ -category, it suffices to show that the monoidalproduct preserves colimits in the second entry. This follows from the observation that colimits in { A , . . . , A n } are computed pointwise, i.e. the n restriction functors { A , . . . , A n } → A i preservecolimits.We also introduce the following notation used later on. Notation 2.31.
Let p : Γ → ∆ n be an inner fibration. Given an edge e : a → a ′ in Γ we write e : a ! −→ a ′ if e is a p -coCartesian edge and e : a ∗ −→ a ′ if e is a p -Cartesian edge.Our next goal is to describe an analogue of the construction of the semiorthogonal decom-position in Lemma 2.30 in the setting of dg-categories and show that the resulting ∞ -categorieswith semiorthogonal decompositions are equivalent. For that we recall the notion of gluingfunctors of semiorthogonal decompositions of length 2, see also [DKSS19]. Definition 2.32.
Let V be a stable ∞ -category with a semiorthogonal decomposition ( A , B ).We define a simplicial set χ ( A , B ) by defining an n -simplex of χ ( A , B ) to correspond to thefollowing data.• An n -simplex j : ∆ n → ∆ of ∆ .• An n -simplex σ : ∆ n → V such that σ (∆ j − (0) ) ⊂ A and σ (∆ j − (1) ) ⊂ B .We define the face and degeneracy maps to act on an n -simplex ( j, σ ) ∈ χ ( A , B ) n componentwise.We denote by p : χ ( A , B ) → ∆ the apparent forgetful functor. Definition 2.33.
Let V be a stable ∞ -category with a semiorthogonal decomposition ( A , B ).We call• ( A , B ) Cartesian if the functor p : χ ( A , B ) → ∆ is a Cartesian fibration. In that case, wecall the functor associated to the Cartesian fibration p the right gluing functor associatedto ( A , B ). ( A , B ) coCartesian if the functor p : χ ( A , B ) → ∆ is a coCartesian fibration. In thatcase, we call the functor associated to the coCartesian fibration p the left gluing functorassociated to ( A , B ). Lemma 2.34 ([DKSS19]) . Let V be a stable ∞ -category with a semiorthogonal decomposition ( A , B ) . • If ( A , B ) is Cartesian, the inclusion functor A → V admits a right adjoint, the restrictionof which to B is the right gluing functor of ( A , B ) . • If ( A , B ) is coCartesian, the inclusion functor B → V admits a left adjoint, the restrictionof which to A is the left gluing functor of ( A , B ) . The next proposition can be summarized as showing that Cartesian semiorthogonal decom-positions of length 2 are fully determined by their left gluing functor and dually that coCartesiansemiorthogonal decomposition of length 2 are fully determined by their right gluing functor.
Proposition 2.35 ([DKSS19]) . Let V be a stable ∞ -category with a semiorthogonal decompo-sition ( A , B ) . • If ( A , B ) is Cartesian with right gluing functor G , there exists an equivalence of ∞ -categories V ≃ Fun ∆ (∆ , χ ( G )) , where χ ( G ) → ∆ is the Cartesian fibration classifying G considered as a functor ∆ → Cat ∞ . • If ( A , B ) is coCartesian with left gluing functor F , there exists an equivalence of ∞ -categories V ≃ Fun ∆ (∆ , Γ( F )) . We now generalize the notion of gluing functor to semiorthogonal decompositions of arbitrarylength and show that such semiorthogonal decompositions are also determined by their gluingfunctors.
Definition 2.36.
Let V be a stable ∞ -category with a semiorthogonal decomposition ( A , . . . , A n ).We call• ( A , . . . , A n ) a Cartesian semiorthogonal decomposition if each semiorthogonal decompo-sition ( A i , ⊥ A i ) is Cartesian. In that case, we call the right gluing functor of ( A i , ⊥ A i )the i th right gluing functor of ( A , . . . , A n ).• ( A , . . . , A n ) a coCartesian semiorthogonal decomposition if each semiorthogonal decom-position ( A i , ⊥ A i ) is coCartesian. If ( A , . . . , A n ) is coCartesian, we call the left gluingfunctor of ( A i , ⊥ A i ) the i th left gluing functor of ( A , . . . , A n ).We now introduce a dg-analogue of Lemma 2.30: semiorthogonal decompositions arisingfrom upper triangular dg-algebras concentrated on the diagonal and upper minor diagonal. Definition 2.37.
For 1 ≤ i ≤ n , let A i be a dg-algebra and for 1 ≤ i ≤ n − M i be an A i - A i +1 -bimodule. We denote by A = A M . . . A M . . . A . . . . . . A n be the upper triangular dg-algebra, i.e. the dg-algebra with underlying chain complex M ≤ i ≤ n A i ⊕ M ≤ i ≤ n − M i nd multiplication · given by a i · a ′ j = δ i,j a i a ′ j , m i · m ′ j = 0 ,a i · m j = δ i,j a i .m j , m j · a i = δ j +1 ,i m j .a i , where a i ∈ A i , a ′ j ∈ A j and m i ∈ M i , m ′ j ∈ M j and δ i,j denotes the Kronecker delta. Proposition 2.38.
Let A be an upper triangular dg-algebra as in Definition 2.37. Then thestable ∞ -category D ( A ) carries a semiorthogonal decomposition ( D ( A ) , . . . , D ( A n )) of length n with i th left gluing functor - ⊗ A i M i .Proof. The upper triangular dg-algebra A is quasi-isomorphic to the upper triangular dg-algebra obtained from replacing M i by a cofibrant replacement. We thus assume without lossof generality that the M i are cofibrant bimodules. Consider the morphisms of dg-algebras v i : A i → A and w i : A → A i , given on the underlying chain complexes by the inclusion ofthe direct summand A i and the projection to the summand A i , respectively. The dg-functor v i ! = - ⊗ dg A i A : dgMod( A i ) → dgMod( A ) and the pullback ( w i ) ∗ determine right A -modules v i ! ( A i )and ( w i ) ∗ ( A i ) with underlying chain complexes A i ⊕ M i , where we set M n = 0, and A i , respec-tively. The functors ψ ( v i ! ) and ψ (( w i ) ∗ ) both exhibit D ( A i ) ⊂ D ( A ) as a stable subcategory. Forconcreteness, we denote the stable subcategories obtained from ψ ( v i ! ) by D ( A i ) v and the stablesubcategories obtained from ψ (( w i ) ∗ ) by D ( A i ) w . We wish to show that ( D ( A ) v , . . . , D ( A n ) v )is a semiorthogonal decomposition of D ( A ). For that it suffices to show statements i ) and ii )of Lemma 2.28. To show statement ii ), it suffices to show conditions 1 and 2 of Lemma 2.29 forthe pairs of stable subcategories D ( A i ) v , h D ( A i +1 ) v , . . . , D ( A n ) v i ⊂ h D ( A i ) v , . . . , D ( A n ) v i forall 1 ≤ i ≤ n .We compute for an A i -module N i and an A j -modules N j the Hom complexHom dgMod( A ) ( v i ! ( N i ) , v j ! ( N j )) ≃ Hom dgMod( A i ) ( N i , N j ) if i = j, Hom dgMod( A j ) ( N i ⊗ A i M i , N j ) if i + 1 = j, . (15)This shows condition 1 of Lemma 2.29.We observe that the datum of a right dg-module N over A is equivalent to the datum of asequence N f −−→ N f −−→ . . . f n − −−−→ N n where N i is a right A i ≃ End dg (( w i ) ∗ ( A i ))-module and f i ∈ M i ( N i , N i +1 ). Denote by N ≥ i the submodule N i f i −→ . . . f n − −−−→ N n of N . We thus find distinguished triangles N ≥ i +1 → N ≥ i → N i in dgMod( A ). As shown in [Fao17, 4.3.1], the image under the dg-nerve of adistinguished triangle in a dg-category can be extended to a fiber and cofiber sequence. Wecan thus express N ∈ D ( A ) as repeated cofibers of modules N i ∈ D ( A i ) w ⊂ D ( A ) with1 ≤ i ≤ n . Clearly N n ∈ D ( A n ) w = D ( A n ) v . A simple induction, using that there existdistinguished triangles in dgMod( A ) of the form N i → N i ⊗ A i v i ! ( A i ) → N i ⊗ A i M i , thus showsthat N ∈ h D ( A ) v , . . . , D ( A n ) v i . It follows that statement i ) of Lemma 2.28 is fulfilled.Consider the subalgebra A ≥ i of A with underlying chain complex M i ≤ k ≤ n A k ⊕ M i ≤ k ≤ n − M k . The fully faithful dg-functor dgMod( A ≥ i ) → dgMod( A ) induces a fully faithful functor of ∞ -categories ι : D ( A ≥ i ) → D ( A ). The above arguments show that the essential image of ι is h A i , . . . , A n i . The above arguments can thus easily be adapted to also show condition 2 of emma 2.29. We have thus proven the existence of the desired semiorthogonal decompositionof D ( A ).We now determine the i th left gluing functor of ( D ( A ) v , . . . , D ( A n ) v ). Consider the fully-faithful left Quillen functor- ⊗ dg A i A i : dgMod( A i ) → dgMod( A ≥ i ) . The right adjoint is given by the Quillen functor Hom dgMod ( A ≥ i ) ( A i , -), the restriction of whichto dgMod( A ≥ i +1 ) is given by Hom dgMod( A ≥ i +1 ) ( M i , -), which in turn is left adjoint to - ⊗ dg A i M i .Passing to the underlying adjunctions of ∞ -categories of the above Quillen adjunctions showsthat the i th left gluing functor of ( D ( A ) , . . . , D ( A n )) is given by - ⊗ A i M i . Remark 2.39.
An illuminating discussion of the role of the morphism of dg-algebra v i and w i appearing in the proof of Proposition 2.38 and the resulting stable subcategories of D ( A ) canbe found in [Bar20, Section 2.3.2]. Proposition 2.40.
For ≤ i ≤ n , let A i be a dg-algebra and for ≤ i ≤ n − , let M i be an A i - A i +1 -bimodule. Denote by A the upper triangular dg-algebra of Definition 2.37. Considerthe diagram α : ∆ n − → LinCat k corresponding to D ( A ) ⊗ A M −−−−−→ D ( A ) ⊗ A M −−−−−→ . . . ⊗ An − M n − −−−−−−−−−→ D ( A n ) . Then there exists an equivalence of ∞ -categories D ( A ) ≃ { D ( A ) , . . . , D ( A n ) } such that for all ≤ i ≤ n , the following diagram commutes. D ( A i ) D ( A ) { D ( A ) , . . . , D ( A n ) } ψ ( v i ! )[ n − i ] ι i ≃ (16) Proof.
Using that the left gluing functors of the semiorthogonal decompositions ( D ( A ) , . . . , D ( A n ))of { D ( A ) , . . . , D ( A n ) } and D ( A ) are equivalent, it follows from a repeated application of Propo-sition 2.35 that there exists an equivalence of ∞ -categories D ( A ) ≃ { D ( A ) , . . . , D ( A n ) } . Theobservation that (16) commute, follows from the observation that the equivalences of Proposi-tion 2.35 commute with the inclusion functors of the components of the semiorthogonal decom-position, up to delooping. Notation 2.41.
Let A be an upper triangular dg-algebra as in Definition 2.37 and 1 ≤ i ≤ n .Using the notation from the proof of Proposition 2.38, we denote by p i A = ψ ( v i ! )( A i ) ∈ D ( A ). Perverse sheaves have their origin in a homology theory of stratified topological spaces calledintersection homology. A perverse sheaf is an object in the category of constructible sheaves, i.e.the category of complexes of sheaves which are locally constant on any stratum. The homologyof the stratified space is obtained via the sheaf cohomology of the perverse sheaf. Perversityof a constructible sheaf is a condition on its homology with and without compact support.For example, perversity of constructible sheaves on a complex surface implies that they areconcentrated in degree 0 away from their singularities and concentrated in degrees 0 , n some nice stratified spaces, there are known descriptions of the abelian category ofperverse sheaves in terms of quiver representations, see e.g. [KS16b] for an overview. A way toobtain such a description, is to identify a suitable ’skeleton’ of the stratified space and describesthe perverse sheaf in terms of certain homology groups with support restrictions related to theskeleton. These homology groups have to fulfil the crucial restriction that they are concentratedin a single degree. The most iconic such description is of the abelian category of perverse sheaveson a disc with a singularity in the center, in terms of the category of diagrams of vector spaces r : V ↔ N : s such that r s − id N and s r − id V are equivalences. The vector spaces N and V are called nearby and vanishing cycles, respectively.While it is currently not clear how to categorify constructible sheaves and thus perversesheaves directly, the remarkable idea of [KS14] is to categorify perverse sheaves using theirquiver descriptions, when available. The ’ad-hoc’ categorification proposed in [KS14] of theabove mentioned quiver description of perverse sheaves on a disc is a spherical adjunction.Further descriptions of the category of perverse sheaves on a disc with a singularity inthe center in terms of quiver representations were given in [KS16a]. For each n ≥
2, thecategory of perverse sheaves is equivalent to the abelian category of diagrams of vector spaces r i : V n ↔ N i : s i for 1 ≤ i ≤ n , such that• r i ◦ s i = id N i ,• r i +1 ◦ s i (with i + 1 modulo n ) is an isomorphism for 1 ≤ i ≤ n • and r i ◦ s j = 0 for i = j, j + 1 mod n .The vector spaces N i are all equivalent, they are the nearby cycles. The vector space V n corresponds to the vector space of sections of the perverse sheaves with support on n outgoingrays starting at the origin, see Figure 1. n outgoing rays with a chosen order. The map r i is defined as the restriction map to a point on the i th outgoing ray. Notethat the restriction maps r i have a paracyclic symmetry, meaning a cyclic symmetry up tothe monodromy of the perverse sheaf, arising from the cyclic symmetry of the n rays given byrotating the disc by 2 π/n . In Section 3.1 we describe an ad-hoc categorification of this quiverdescription, which will provide a local description of a parametrized perverse schober. Thecategorification is based on Dyckerhoff’s categorified Dold-Kan correspondence, see [Dyc17]. Asnoted in loc. cit. , the relevance for the categorification of the local description of perverse sheaveswas one of the motivations for the inception of the categorified Dold-Kan correspondence. We begin with briefly recalling the concept of a spherical adjunction. Consider an adjunctionof stable ∞ -categories F : A ↔ B : G . We associate the following endofunctors. The twist functor T A is defined as the cofiber in the stable ∞ -category Fun( A , A ) of theunit map id A → GF of the adjunction F ⊣ G .• The cotwist functor T D is defined as the fiber in the stable ∞ -category Fun( B , B ) of thecounit map F G → id B of the adjunction F ⊣ G .The adjunction F ⊣ G is called spherical if the functors T A and T B are equivalences. In thiscase, the functor F is also called spherical. A spherical functor F admits repeated left and rightadjoints, each given by the composite of F or G with a power of the twist or cotwist functor. Fora treatment of spherical adjunctions in the setting of stable ∞ -categories, we refer to [DKSS19]and [Chr20].A 2-simplicial stable ∞ -category is an ( ∞ , (op , -) → S t , from the 2-categoricalversion of the simplex category to the ( ∞ , S t of the ∞ -category St of stable ∞ -categories. The categorified Dold-Kan correspondence of [Dyc17] is an adjoint equivalence be-tween the ∞ -category of bounded below complexes of stable ∞ -categories and the ∞ -categoryof 2-simplicial stable ∞ -categories. The right adjoint is called the categorified Dold-Kan nerve N . The categorified Dold-Kan nerve N generalizes the well known construction from K -theorycalled the Waldhausen S • -construction. More precisely, given a complex of stable ∞ -categoriesconcentrated in degrees 0 ,
1, the categorified Dold-Kan nerve recovers Waldhausen’s relative S • -construction. We refer to [Dyc17] for further details.Let F : A ↔ B be a spherical adjunction. We consider the spherical functor G : B → A as acomplex of stable ∞ -categories concentrated in degrees ,
1, denoted G [0]. We further denoteby B [1] the complex concentrated in degree 1 with value B . Consider the morphism betweencomplexes of stable ∞ -categories G [0] → B [1], depicted as follows. A B . . . B . . . G id B Applying the categorified Dold-Kan nerve N , we obtain a morphism φ ∗ : N ( G [0]) ∗ → N ( B [1]) ∗ between the simplicial objects in St underlying the 2-simplicial objects in S t . Spelling out thedefinition of the categorified Dold-Kan nerve and using a property of Kan extensions, namely[Lur09, 4.3.2.15], we obtain the following. Lemma 3.1.
Let F : A ↔ B : G be a spherical adjunction and N ( G [0]) ∗ and N ( B [1]) ∗ as above.There exist the following equivalences between ∞ -categories.1. N ( G [0]) ≃ A .2. N ( G [0]) n ≃ { A , B , . . . , B } for n ≥ , in the notation of Lemma 2.30, corresponding to thefollowing sequence of n functors. A F −−→ B id −−→ B id −−→ . . . id −−→ B N ( B [1]) ≃ B . We propose that for n ≥
0, the ∞ -category N ( G [0]) n of n -simplicies categorifies the vectorspace V n +1 of sections supported on n + 1-outgoing rays and the ∞ -category N ( B [1]) of 1-simplicies categorifies the vector spaces N i of nearby cycles. We use the homological indexing convention for all complexes in this paper. otation 3.2. Let F ⊣ G be as above. We denote• V F = A .• V nF = { A , B , . . . , B | {z } n − } for n ≥ N F = B .Assume that n ≥
3. We propose that the first restriction map r : V n → N is categorifiedby the functor ̺ : V nF ≃ N ( G [0]) n − d −→ N ( G [0]) n − d −→ . . . d −→ N ( G [0]) φ −→ N ( B [1]) ≃ N F obtained from composing φ with repeated 0th face maps of the simplicial structure of N ( G [0]) ∗ .The functor ̺ can equivalently be described as the projection functor π n to the n th componentof the semiorthogonal decomposition ( A , B , . . . , B ) of length n of V nF . If n = 1, we propose thatthe restriction map r is categorified by F : A → B and if n = 2 we propose that the restrictionmap r is categorified by φ = π . To categorify the further restriction maps, we need to takeinto account the paracyclic symmetry. The description of the categorification of V n in terms of V nF however obscures this paracyclic symmetry. One way to solve this is by lifting the simplicialobject N ( G [0]) ∗ to a paracyclic object. This approach is realized in [DKSS19]. One can thenreplace V nF by an equivalent ∞ -category where the paracyclic symmetry is apparent. For nowwe adopt a more pedestrian approach and simply require that there be a sequence of adjunctions ̺ n ⊣ ς n ⊣ ̺ n − ⊣ · · · ⊣ ς ⊣ ̺ ⊣ ς , (17)where ̺ is as above and propose that ς i categorifies s i and ̺ i categorifies r i . We describe theparacyclic symmetry of V nF and justify this proposal in Section 2.2. We call the ̺ i the categorifiedrestriction maps. A direct computation shows that the functors ̺ i and ς i are described as follows. Lemma 3.3.
Let F ⊣ G as above and n ≥ . Consider the following functors ̺ i : V nF → N F and ς i : N F → V nF for ≤ i ≤ n . • If n = 1 , we set ̺ = F and ς = G . • If n ≥ , we set ̺ i = π n for i = 1 , fib n − i − ,n − i [ i − for ≤ i ≤ n − , rfib , [ n − for i = n. The functor rfib , denotes the composition of the projection functor to the first two com-ponents of the semiorthogonal decomposition with the relative fiber functor that assigns toa vertex a → b ∈ { A , B } the vertex fib( F ( a ) → b ) ∈ B . The functor fib i − ,i [ n − i ] denotesthe composition of the projection functor to the ( i − th and i th component with the fiberfunctor. • If n ≥ , we set ς to be the functor that assigns to b ∈ B the object G ( b ) ∗ −→ b id −→ . . . id −→ b in V nF and set for ≤ i ≤ n ς i = ( ι B ) n − i +2 [ − i + 2] , where ( ι B ) j is the inclusion of the j th component of the semiorthogonal decomposition.These functors form the sequence of adjunctions (17) . We are now ready to describe the local model for a parametrized perverse schober at asingular vertex of valency n . efinition 3.4. Let F : A ↔ B : G be an adjunction of stable ∞ -categories and n ≥ C n = ( { , . . . , n } ) ⊳ . If n = 1, we denote by G ( F ) : C ≃ ∆ → St thefunctor F . If n ≥
2, we denote by G n ( F ) the functor C n → St assigning• to the initial vertex ∗ ∈ C n the stable ∞ -category V nF ,• to each vertex i ∈ C n the stable ∞ -category N F ,• to each edge ∗ → i the functor ̺ i from Lemma 3.3.The adjoint functors ς i will feature in the local description of duals of parametrized perverseschobers, see Section 4.3. We begin by recalling the definition of the paracyclic 1-category Λ ∞ . Definition 3.5.
For n ≥
0, let [ n ] denote the set { , . . . , n } . The objects of Λ ∞ are the sets [ n ].The morphism in Λ ∞ are generated by morphisms• δ , . . . , δ n : [ n − → [ n ],• σ , . . . , σ n − : [ n ] → [ n − τ n,i : [ n ] → [ n ] with i ∈ Z subject to the simplicial relations and the further relations τ n,i ◦ τ n,j = τ n,i + j , τ n, = id [ n ] ,τ n, δ i = δ i − τ n − , for i > , τ n, δ = δ n ,τ n, σ i = τ n +1 , σ i − for i > , τ n, σ = σ n τ n +1 , . The simplex category ∆ is a subcategory of Λ ∞ . A paracyclic object in an ∞ -category C is a functor Λ op ∞ → C , where we identify the 1-category Λ op ∞ with its nerve. A paracyclic objectin C is thus a simplicial object X ∗ ∈ Fun(∆ op , C ) with face maps d i and degeneracy maps s i together with a sequence of paracyclic isomorphisms t n : X n → X n satisfying d i t n = t n − d i − for i > , d t n = d n and (18) s i t n = t n +1 s i − for i > , s t n = t n +1 s n . (19)Let F ⊣ G be a spherical adjunction. As shown in [DKSS19], the simplicial object N ( G [0]) ∗ can be lifted to a paracyclic object. We emphasize that the sphericalness of the adjunction F ⊣ G is crucial for showing that the paracyclic isomorphism t n of this paracyclic structureis really an isomorphism. In this section we give an alternative description of the paracyclicisomorphisms t n in terms of the twist functor T V nF of a spherical adjunction F ′ ⊣ G ′ describedbelow in Lemma 3.8. We then proceed to show that this isomorphisms realizes the paracyclicsymmetry of the functors ̺ i and ς i . Construction 3.6.
Let F : A ↔ B : G be a spherical adjunction. Denote the left adjoint of F by E . Consider the full subcategory M of the ∞ -category of diagrams Fun(∆ × ∆ , Γ( F )) ofthe form a a ′ b ′ b ! ∗ The left cone ( { , . . . , n } ) ⊳ is defined as the simplicial join ∆ ∗ { , . . . , n } . ith a, a ′ ∈ A and b, b ′ ∈ B . The restriction functor res : M → { A , B } , given by the projectionto the edge a → b is a trivial fibration. As shown in [DKSS19], it follows from the sphericalnessof the adjunction F ⊣ G , that the fiber functor in the horizontal direction M → { A , B } isalso an equivalence. By choosing a section of the trivial fibration res and composing with thefiber functor we obtain an autoequivalence τ : { A , B } → { A , B } , called the relative suspensionfunctor in loc. cit . Lemma 3.7.
Let F : A ↔ B : G be a spherical adjunction with cotwist functor T B . Denote theleft adjoint of F by E . The left adjoint of the functor V F = { A , B } rfib −−→ B (20) is given by the functor that assigns E ( b ) ∗ −→ T − B ( b ) to b ∈ B .Proof. As shown in [Chr20, 1.31], the stable subcategories A ⊥ , A , B , ⊥ B ⊂ { A , B } form semiorthog-onal decompositions ( A ⊥ , A ) , ( A , B ) , ( B , ⊥ B ) of { A , B } . We denote by i B , i A ⊥ the inclusionfunctors of B and B ≃ A ⊥ into { A , B } , respectively. The functor i A ⊥ assigns to b ∈ B ≃ A ⊥ the object G ( b ) ! −→ b ∈ { A , B } . It is easily checked that there is a sequence of adjunctionsrfib[1] ⊣ i B ⊣ π ⊣ i A ⊥ . (21)Composing with the adjunction τ − ⊣ τ , where τ is the relative suspension functor from Con-struction 3.6, with the sequence of adjunction (21) yields the sequence of adjunctions π [1] ⊣ i A ⊥ [ − ⊣ T − B rfib[1] ⊣ i B T B . We have thus established the desired adjunction i A ⊥ T − B ⊣ rfib. Lemma 3.8.
Let F : A ↔ B : G be a spherical adjunction with cotwist functor T B . Denote theleft adjoint of F by E . For n ≥ , consider the functor F ′ : V nF −→ N × nF with components F ′ = ( ̺ , . . . , ̺ n ) .1. The functor F ′ admits left and right adjoints E ′ , respectively, G ′ , given by E ′ = ( ς , . . . , ς n , ς T − B [1 − n ]) ,G ′ = ( ς , . . . , ς n ) .
2. The adjunction F ′ ⊣ G ′ is spherical.Proof. We begin with showing statement 1. The adjunction F ′ ⊣ G ′ follows from composingthe adjunctions ̺ i ⊣ ς i with the adjunction ∆ ⊣ ⊕ between the constant diagram functor∆ : N F → N ⊗ nF and its right adjoint given by the direct sum functor. Again by composingadjunctions, we obtain that to show that E ′ is left adjoint to F ′ it suffices to show that ς T − B [ n ]is left adjoint to ̺ n . This follows directly from the following observations.• The functor ̺ n factors as V nF π , −−→ V F rfib[ n − −−−−−−→ N F . • The left adjoint of rfib : V F → N F was determined in Lemma 3.7 and is given by thefunctor that maps b ∈ N F to E ( b ) ∗ −→ T − B ( b ).• The left adjoint of π , is given by the functor that maps E ( b ) ∗ −→ T − B ( b ) ∈ V F to E ( b ) ∗ −→ T − B ( b ) id −→ . . . id −→ T − B ( b ) ∈ V nF . or statement 2, consider the endofunctor M = F ′ G ′ : N × nF → N × nF of the adjunction F ′ ⊣ G ′ . We can depict M as the following matrix. id N F id N F . . . N F id N F . . . N F . . . . . . id N F id N F T B [ n −
1] 0 0 . . . N F The counit cu : M → id N × nF is the projection to the diagonal, so that we deduce that thecotwist T N × nF is an equivalence. We further observe that there exists an equivalence cu ◦ T N × nF ≃ T N × nF ◦ cu . The left adjoint E ′ : N × nF → V nF clearly satisfies G ′ ◦ T − N × nF . We have shown that allconditions of [Chr20, 6.7] are fulfilled and it follows that the adjunction F ′ ⊣ G ′ spherical. Remark 3.9.
We highlight the relation of Lemma 3.8 to other results in the literature. Let F ⊣ G be a spherical adjunction. Consider further the (trivially) spherical adjunction 0 B :0 ↔ B : 0 ′ B and denote by F ′′ : V n B ↔ B n +1 : G ′′ the spherical adjunction associated inLemma 3.8 to 0 B ⊣ ′ B . The adjunction F ′′ ⊣ G ′′ appears in the special case B = D ( k ) perf in [BD19, 5.14], where it is shown that F ′′ carries a left Calabi-Yau structure. The sphericaladjunction F ′ ⊣ G ′ associated to F ⊣ G in Lemma 3.8 can be described as the composition ofthe spherical adjunctions F ′′ ⊣ G ′′ and( F, . . . ,
0) : A ←→ B n +1 : ( G, , . . . , Remark 3.10.
Consider the setting of Lemma 3.8. Lurie’s ∞ -categorical Barr-Beck theoremimplies that the adjunction F ′ ⊣ G ′ is monadic. Further, if the adjunction F ⊣ G is monadic,then the adjunction F ′ ⊣ G ′ is also comonadic. Lemma 3.7 thus implies that a monadic sphericaladjunction F ⊣ G can be recovered from the comonad M = F ′ G ′ : N × nF → N × nF whose underly-ing endofunctor is determined by the twist functor T B . This does not imply that the sphericalmonadic adjunction F ⊣ G can be recovered from its twist functor, see also [Chr20, Section 4.1].All further data is encoded in the comonad structure of M . Proposition 3.11.
Let F : A ↔ B : G be a spherical adjunctions and consider the twist functor T V nF of the spherical adjunction F ′ ⊣ G ′ described in Lemma 3.8. Then there exist equivalencesof functors ̺ i ◦ T V nF = ( ̺ i +1 for ≤ i ≤ n − T B [ n − ◦ ̺ for i = n (22) and T − V nF ◦ ς i = ( ς i +1 for ≤ i ≤ n − ς ◦ T − B [1 − n ] for i = n (23) Proof.
By the 2/4 property of spherical adjunctions there exists an equivalence T − V nF G ′ ≃ E ′ ,showing the identities (23). The identities (22) follow from passing to left adjoints. Parametrized perverse schobers globally
In this section we discuss ribbon graphs and their relation to triangulated surfaces, similar tothe treatments in [DK15, DK18]. The only difference is that we consider ribbon graphs withadditional decorations called singularities . Definition 4.1. • A graph Γ consists of two finite sets Γ of vertices and H Γ of halfedges (sometimes simplycalled H) together with an involution τ : H → H and a map σ : H → Γ .• Let Γ be a graph. We denote by Γ the set of orbits of τ . The elements of Γ are calledthe edges of Γ. An edge is called internal if the orbit contains two elements and calledexternal if the orbit contains a single element. We denote the set of internal edges of Γ byΓ ◦ and the set of external edges by Γ ∂ .• A graph with singularities (Γ , V ) consists of a graph Γ together with a subset V ⊂ Γ ofvertices of Γ called singularities.• A ribbon graph with singularities consists of a graph with singularities (Γ , V ) togetherwith a choice of a cyclic order on the set H( v ) of halfedges incident to v for all v ∈ Γ . Remark 4.2.
A total order on a finite set H with cardinality n can be defined as a bijection φ : { , . . . , n } ≃ H . Such a total order induces a cyclic order, where φ ( i + 1) follows φ ( i ) if i = n and φ (1) follows φ ( n ). Definition 4.3.
Let Γ be a graph. We denote by Exit(Γ) the poset with• the set of elements Γ ∐ Γ and• all non-identity morphisms of the form v → e with v ∈ Γ a vertex and e ∈ Γ an edgeincident to v .We call Exit(Γ) the exit path category of Γ. Remark 4.4.
The exit path category Exit(Γ) is in the literature also referred to as the incidencecategory I (Γ).By a surface we mean a smooth surface with possibly empty boundary. Remark 4.5.
Let (Γ , V ) be a graph with singularities and S an oriented surface. Consider anembedding of the geometric realization | Exit(Γ) | of the exit path category into S . This embed-ding determines two canonical ribbon graphs with singularities (Γ , V ), where the orientation ofthe halfedges at any vertex in a local chart of S is either clockwise or counter-clockwise. We willexclusively consider the ribbon graph with the counter-clockwise orientations, to later matchthe orientation of the quivers inscribed into ideal triangulations of surfaces. Notation 4.6.
We use a graphical notation for ribbon graphs with singularities. We denotenon-singular vertices by · , singular vertices by × and internal edges by a straight line. Wedenote external edges as follows. · Example 4.7.
The following diagram ×· · enotes a ribbon graph with singularities (Γ , V ) with | Γ | = 3, one singular vertex, i.e. | V | = 1,one external edge and one loop and the cyclic order at each vertex induced by the counter-clockwise orientation of the plane. Definition 4.8.
Let S be a marked surface, i.e. equipped with a finite subset M of S of markedpoints. A simple curve in S is a curve in S such that• the endpoints of γ lie in M ,• γ does not intersect M and ∂ S , except at the endpoints,• γ does not self-intersect, except that its endpoints may be coincide,• if γ is a closed loop, then γ is not contractible into M or ∂ S .An arc in S is an equivalence class of curves under isotopy and reversal of parametrization. Twoarcs are called compatible if there are curves in their respective isotopy classes which do notintersect, except possibly at the endpoints. Definition 4.9 ([FST08, 2.6]) . Let S be a marked surface. An ideal triangulation of S consistsof a maximal collection of distinct pairwise compatible arcs in S .Any collection of distinct and pairwise compatible arcs can be realized by curves in respectiveisotopy classes which do not intersect except for the endpoints, see [FST08, 2.5]. Given an idealtriangulation T of a surface S , we choose such a collection of non-intersecting curves. Thesecurves cut S into ideal triangles whose vertices lie in M . Note that triangles may be self-folded,i.e. two of the sides of the triangle are identical. Up to two sides of an ideal triangle may lie onthe boundary and be given by non-simple curves in ∂ S connecting two marked points. We callthe elements of the collection of non-intersecting simple curves interior edges of T and the sidesof the ideal triangles given by non-simple curves on the boundary the boundary edges of T .Each ideal triangulation of an oriented surface determines a ribbon graph with singularities(Γ T , V T ). Definition 4.10.
Let S be an oriented surface with an ideal triangulation T . We denote by(Γ T , V T ) the ribbon graph with singularities determined by the following.• The set of vertices of Γ T is the set of ideal triangles of T .• All vertices are singular, i.e. V T = (Γ T ) .• The set of internal edges of Γ T is the set of interior edges of T . An internal edge e represented by an edges γ e is incident to the vertices of Γ T corresponding to the two idealtriangles incident to γ e . Self-folded triangles give rise to internal loops in Γ T .• The set of external edges of Γ T is the set of boundary edges of T . Such an external edgeis incident to the vertex of Γ T corresponding to the ideal triangle which it is a side of.• Given a vertex v of Γ T , the cyclic order of H( v ) is given by the counter-clockwise cyclicorder of the edges of the corresponding ideal triangle of T .We call (Γ T , V T ) the dual ribbon graph with singularities of T . Example 4.11.
Consider the following two ideal triangulations of the once-punctured disc. he dual ribbon graphs with singularities can be depicted as follows. ×× × × Ribbon graphs with singularities can be glued along their external edges.
Construction 4.12.
Let (Γ ′ , V ′ ) and (Γ ′′ , V ′′ ) be ribbon graphs with singularities and let I bea finite set and i ′ : I → (Γ ′ ) ∂ and i ′′ : I → (Γ ′′ ) ∂ injective maps. Then there exists a ribbongraph with singularities (Γ , V ) determined by• Γ = Γ ′ ∪ Γ ′′ ,• Γ = (cid:0) Γ ′ \ i ′ ( I ) (cid:1) ∪ (cid:0) Γ ′′ \ i ′′ ( I ) (cid:1) ∪ I ,• V = V ′ ∪ V ′′ ,• the cyclic order of H( v ) of a vertex v ∈ Γ ′ ⊂ Γ and v ∈ Γ ′′ ⊂ Γ being given by the cyclicorder of H( v ) determined by the ribbon graph (Γ ′ , V ′ ), respectively, (Γ ′′ , V ′′ ).We call (Γ , V ) the gluing of (Γ ′ , V ′ ) and (Γ ′′ , V ′′ ) along I . Note that there exists an equivalenceof posets Exit(Γ) ≃ Exit(Γ ′ ) ∐ I Exit(Γ ′′ ). Example 4.13.
Let T be an ideal triangulation of a surface. For each ideal triangle F i weassociate a ribbon graph with singularities (Γ i , V i ) as follows.• If F i is not a self-folded triangle, then (Γ i , V i ) is the following ribbon graph with singular-ities. × • If F i is a self-folded triangle, then (Γ i , V i ) is the following ribbon graph with singularities. × Then (Γ T , V T ) is the gluing of the ribbon graphs with singularities (Γ i , V i ) along their externaledges determined by the incidence of the ideal triangles. We essentially define a parametrized perverse schober as a collection of the local models ofDefinition 3.4 arising from spherical adjunctions which are suitably glued together along aribbon graph with singularities.
Definition 4.14.
A perverse schober parametrized by a ribbon graph with singularities (Γ , V )is defined to be a functor F : Exit(Γ) → St satisfying the following.1. For each vertex p ∈ Γ there exists a spherical functor F : A → B and a choice ofequivalence of posets C n ≃ Exit(Γ) p/ , respecting the cyclic ordering of { , . . . , n } andH( v ), such that the restriction of F to the ribbon corolla C n ≃ Exit(Γ) p/ is equivalent to G n ( F ) as objects in Fun( C n , St).2. If p ∈ Γ \ V and F : A → B the corresponding functor from condition 1, then A ≃ he ∞ -category P (Γ , V ) of (Γ , V )-parametrized perverse schobers is defined to be the fullsubcategory of the functor category Fun(Exit(Γ) , St) spanned by perverse schobers.
Notation 4.15.
We will use a graphical notation for perverse schobers parametrized by ribbongraphs similar to the graphical notation for ribbon graphs introduced in Notation 4.6. Wedenote a parametrized perverse schober by specifying the spherical functor at each vertex of thecorresponding ribbon graph and specifying the functor associated to each non-identity morphismin the exit path category.
Example 4.16.
Let F : A → B be a spherical functor and T : B → B some autoequivalence.The diagram F B B ( T ◦ ̺ ,̺ )( ̺ ,̺ )( ̺ ,̺ ) ̺ (24)corresponds to the parametrized perverse schober given by the following Exit(Γ)-indexed dia-gram in St. ABB { B , B } B { B , B } B T ◦ F̺ ̺ ̺ ̺ ̺ ̺ The next lemma shows that parametrized perverse schobers can be glued along stops.
Lemma 4.17.
Let (Γ ′ , V ′ ) and (Γ ′′ , V ′′ ) be ribbon graphs with singularities, I a finite set and i ′ : I → (Γ ′ ) ∂ and i ′′ : I → (Γ ′′ ) ∂ injective maps. Denote by (Γ , V ) the glued ribbon graph withsingularities described in Construction 4.12. Consider the functors ev ′ : P (Γ ′ , V ′ ) → Fun( I, St) and ev ′′ : P (Γ ′′ , V ′′ ) → Fun( I, St) , given by the restriction functors along the inclusions I → Exit(Γ ′ ) and I → Exit(Γ ′′ ) , respectively. There exists a pullback diagram in Cat ∞ as follows. P (Γ , V ) P (Γ ′′ , V ′′ ) P (Γ ′ , V ′ ) Fun( I, St) y ev ′′ ev ′ (25) Proof.
Applying the functor Fun(- , St) to the pushout diagram in Cat ∞ I Exit(Γ ′′ )Exit(Γ ′ ) Exit(Γ) p yields the following pullback diagram in Cat ∞ .Fun(Exit(Γ) , St) Fun(Exit(Γ ′′ ) , St)Fun(Exit(Γ ′ ) , St) Fun( I, St) y The statement that the diagram (25) is pullback follows from the following observation: anelement F ∈ Fun(Exit(Γ) , St) lies in P (Γ , V ) if and only if its restriction to Fun(Exit(Γ ′ ) , St)and Fun(Exit(Γ ′′ ) , St) lie in P (Γ ′ , V ′ ) and P (Γ ′′ , V ′′ ), respectively. .3 Global sections and duality Definition 4.18.
Let (Γ , V ) be a ribbon graph with singularities and let F : Exit(Γ) → St bea (Γ , V )-parametrized perverse schober.• We call the stable ∞ -category H (Γ , F ) := lim F the ∞ -category of global sections of F .Global sections form a functor H (Γ , -) : P (Γ , V ) → St . • Let E ⊂ Γ be a subset of edges. Let F be a (Γ , V )-parametrized perverse schober.For e ∈ E , consider the evaluation functor ev e : H (Γ , F ) → F ( e ) contained in the limitdiagram defining H (Γ , F ). We denote by H E (Γ , F ) the full subcategory of H (Γ , F ) spannedby global sections X ∈ H (Γ , F ) such that ev e ( X ) = 0 for all edges e ∈ Γ \ E . We call H E (Γ , F ) the ∞ -category of global sections of F with support on E . If Γ ′ ⊂ Γ is asub-ribbon graph, we also write H Γ ′ (Γ , F ) = H Γ ′ (Γ , F ). Lemma 4.19.
Let (Γ , V ) be a ribbon graph with singularities and F a (Γ , V ) -parametrizedperverse schober. Let Γ ◦ ⊂ E ⊂ Γ and B = E c = Γ \ E . Denote by (Γ ′ , V ) the ribbon graphwith singularities obtained from gluing (Γ , V ) with a ribbon graph with singularities, consistingof a single non-singular vertex and a single external edge, for each edge in B . Denote by F ′ the (Γ ′ , V ) -parametrized perverse schober obtained from gluing F with a copy of the parametrizedperverse schober B for each edge in B . Then there exists an equivalence of ∞ -categories H E (Γ , F ) ≃ H (Γ ′ , F ′ ) . Proof.
The ∞ -categories of global sections H (Γ , F ) and H (Γ ′ , F ′ ) can be described as the ∞ -categories of coCartesian sections of the coCartesian fibrations classified by F : Exit(Γ) → Stand F ′ : Exit(Γ ′ ) → St, respectively. Restriction along Exit(Γ) → Exit(Γ ′ ) induces a functor H (Γ ′ , F ′ ) → H E (Γ , F ) which is an equivalence because it describes left Kan extensions, see[Lur09, 4.2.3.15]. Definition 4.20.
Let (Γ , V ) be a ribbon graph with singularities. We denote Entry(Γ) :=Exit(Γ) op . Given a (Γ , V )-parametrized perverse schober F , we call the right adjoint, respec-tively, left adjoint diagrams D R F , D L F : Entry(Γ) −→ Stthe right dual respectively left dual of F . Remark 4.21.
Consider the setup of Definition 4.20. Lemma 3.8 implies that there existsan equivalence D R F ≃ D L F in Fun(Entry(Γ) , St), which restricts on each singular vertex, withcorresponding spherical adjunction F ⊣ G , to the twist functor of the spherical adjunction F ′ ⊣ G ′ . Remark 4.22.
Let F be a (Γ , V )-parametrized perverse schober. Assume that F : Exit(Γ) → Stfactors through the forgetful functor P r L St → St. The right adjoint diagram D R F : Entry(Γ) → Stthus factors through the forgetful functor P r R → St. There then exists an equivalence of ∞ -categories H (Γ , F ) ≃ colim P r R D R F . (26)Similarly, if F : Exit(Γ) → St factors through the forgetful functor P r R St → St, then there existsan equivalence of ∞ -categories H (Γ , F ) ≃ colim P r L D L F . e can thus, under the assumption of presentability, equivalently express parametrized perverseschobers and their global sections via their duals. These two perspectives may be seen as thecategorified analogue of the two possible perspective on perverse sheaves, either in terms ofsheaves or in terms of cosheaves, see [KS19]. The goal of this section is to show that that contractions of ribbon graphs induce functorsbetween the respective ∞ -categories of parametrized perverse schobers which preserve the globalsections. Definition 4.23. • Let (Γ , V ) be a ribbon graph with singularities and e ∈ Γ an edge connecting two distinctvertices v , v , such that v is not singular. Let { e , e } be the orbit representing the edge e . We define a ribbon graph with singularities (Γ ′ , V ′ ) with – Γ ′ = Γ / ( v ∼ v ) is the set obtained from Γ obtained by identifying v and v , – H Γ ′ = H Γ \{ e , e } , – τ : H Γ ′ → H Γ ′ is the restriction of τ : H Γ → H Γ . – σ : H Γ ′ → Γ ′ is the composite map : H Γ ′ ⊂ H Γ σ −→ Γ → Γ ′ . – V ′ ⊂ Γ ′ consists of the elements of Γ ′ \ [ v ] = Γ \{ v , v } lying in V and v if v issingular, – the cyclic order on H Γ ′ ( v ) with v ∈ Γ ′ \ [ v ] is identical to the cyclic order on H Γ ( v ).Choose any two linear orders of the elements of H Γ ( v ) \{ e } and H Γ ( v ) \{ e } com-patible with the given cyclic ordering. Consider the total order onH Γ ′ ([ v ]) = (cid:0) H Γ ( v ) \{ e } (cid:1) ∪ (cid:0) H Γ ( v ) \{ e } (cid:1) which restricts to the given total orders on H Γ ( v ) \{ e } , H Γ ( v ) \{ e } and such thatall elements of H Γ ( v ) \{ e } follow the elements in H Γ ( v ) \{ e } ). We let the cyclicorder on H Γ ′ ([ v ]) to be the cyclic order induced by the above total order in the senseof Remark 4.2.We call (Γ ′ , V ′ ) the edge contraction of (Γ , V ) at e .• Let (Γ , V ) and (Γ ′ , V ′ ) be ribbon graphs with singularities. We say that there existsa contraction from (Γ , V ) to (Γ ′ , V ′ ) if (Γ ′ , V ′ ) is obtained as a (finitely many times)repeated edge contraction of (Γ , V ). We write c : (Γ , V ) → (Γ ′ , V ′ ). Lemma 4.24.
Let F : A ↔ B : G be a spherical adjunction. Denote by B the functor ofstable ∞ -categories → B . Let m, n ≥ and consider the stable ∞ -categories V m B and V nF withcategorified restriction maps ̺ i : V m B → B , with i = 1 , . . . , m , respectively ̺ j : V nF → B , with j = 1 , . . . , n .1. There exists a pullback diagram in Cat ∞ as follows. V n + m − F V nF V m B B αβ y ̺ ̺ m (27)
2. Denote the categorified restriction maps V n + m − F → B by ̺ , . . . , ̺ n + m − . There existequivalences of functors ̺ j ≃ ̺ j ◦ β and ̺ i + m − ≃ ̺ i ◦ α for j = 1 , . . . , m − and i = 2 , . . . n . roof. Let D : ∆ m − → St be the constant diagram with value B and D : ∆ n − → St, D : ∆ n + m − → St be the diagrams obtained from the sequences of composable functors A G −−→ B id −−→ . . . id −−→ B . The diagram D restricts to the diagrams D and D on ∆ { ,...,n − } and ∆ { n − ,...,n + m − } ,respectively. The inclusion functor ∆ { ,...,n − } ∐ ∆ { n − } ∆ { n − ,...,n + m − } → ∆ n + m − is inneranodyne. It follows that the restriction functorres : Fun ∆ n + m − (∆ n + m − , Γ( D )) → Fun ∆ n − (∆ n − , Γ( D )) × B Fun ∆ m − (∆ m − , Γ( D ))is a trivial fibration. Using the equivalences of ∞ -categories V nF ≃ Fun ∆ n − (∆ n − , Γ( D )) , V m B ≃ Fun ∆ m − (∆ m − , Γ( D )) , V n + m − F ≃ Fun ∆ n + m − (∆ n + m − , Γ( D )) , it follows that there exists a pullback diagram of the form (27). The functors α and β are therestriction functors to the first m − n components, respectively. The description ofthe categorified restriction maps can be checked directly. Construction 4.25.
Consider the setup of Lemma 4.24 and the following diagram, V nF V m B B ̺ j ̺ i (28)where 1 ≤ i ≤ m and 1 ≤ j ≤ n are arbitrary. We can use the autoequivalences (cid:0) T V nF (cid:1) − j and (cid:0) T V m B (cid:1) m − i , defined in terms of the twist functors of the spherical adjunctions described andused in Lemma 3.8 and Proposition 3.11, to find a natural equivalence between the diagram(28) and the following diagram. V nF V m B B ̺ ̺ m (29)In particular, we obtain that the limits of the diagrams (28) and (29) are both equivalentto V n + m − F . Proposition 3.11 also shows that under this equivalence the resulting categorifiedrestriction maps ̺ i : V n + m − F → B are cyclically permuted and may each further change bypost-composition with an autoequivalence of the form ( T B [ n − l for some l ∈ Z . Proposition 4.26.
Let c : (Γ , V ) → (Γ ′ , V ′ ) be a contraction of ribbon graphs with singularities.There is a functor of ∞ -categories c ∗ : P (Γ , V ) → P (Γ ′ , V ′ ) making the following diagramcommute. P (Γ , V ) P (Γ ′ , V ′ )St c ∗ H (Γ , -) H (Γ ′ , -) (30) Proof.
It suffices to show the statement in the case that c is the contraction at an edge e ∈ Γ connecting two vertices v , v such that v is not singular. The edge contraction c induces afunctor Exit( c ) : Exit(Γ) → Exit(Γ ′ ) determined by mapping x ∈ Γ \{ v , v } ⊂ Exit(Γ) to x ∈ Γ \{ v , v } ⊂ Exit(Γ ′ ),• v , v ∈ Γ ⊂ Exit(Γ) to v ,• f ∈ Γ \{ e } ⊂ Exit(Γ) to f ∈ Γ \{ e } ⊂ Exit(Γ ′ ) and• e ∈ Γ ⊂ Exit(Γ) to v ∈ Γ ′ ⊂ Exit(Γ ′ ).We define E to be the poset determined by the following properties.• There exist fully faithful functors Exit(Γ ′ ) , Exit(Γ) → E .• The induced functor Exit(Γ ′ ) ∐ Exit(Γ) → E is bijective on objects• For x ′ ∈ Exit(Γ ′ ) and x ∈ Exit(Γ), there exists a unique morphism from x ′ to x in E if andonly if there exists a morphism x ′ → Exit( c )( x ). There are no morphisms from x to x ′ .Note that the poset E can be equivalently describes as the total space of a Cartesian fibrationclassifying the functor Exit( c ) : ∆ → Cat ∞ .We define c ∗ : Fun(Exit(Γ) , St) → Fun(Exit(Γ) , St) as the composition of the right Kanextension functor along the inclusion Exit(Γ) → E with the restriction functor to Exit(Γ ′ ).It follows from Lemma 4.24 and Construction 4.25 that c ∗ maps P (Γ , V ) to P (Γ ′ , V ′ ). Thecommutativity of the diagram (30) follows from right Kan extensions commuting with rightKan extensions. V nf ∗ This section provides auxiliary computations to be used in Sections 6 and 7. In Section 5.1 westudy the ∞ -category Fun( S n , D ( k )) of local systems on the n -sphere with values in the derived ∞ -category of a commutative ring k and the spherical adjunction f ∗ : D ( k ) ↔ Fun( S n , D ( k )) : f ∗ . We show that there is an equivalence of ∞ -categories Fun( S n , D ( k )) ≃ D ( k [ t n − ]), where k [ t n − ] denotes the polynomial algebra with generator in degree | t n − | = n −
1. In Section 5.2we describe the perverse schober on the disc obtained from the spherical adjunction f ∗ ⊣ f ∗ . In [Chr20], we showed the following.
Proposition 5.1.
For n ≥ let S n denote the singular set of the topological n -sphere andconsider the map f : S n → ∗ . Let further D be a stable ∞ -category and consider the pullbackfunctor f ∗ : D → Fun( S n , D ) (31) with right adjoint f ∗ , given by the limit functor. The adjunction f ∗ ⊣ f ∗ is spherical with twistfunctor T D ≃ [ − n ] . We call the ∞ -category Fun( S n , D ) the ∞ -category of local systems on S n with values in D .It is well known that if D = D ( k ) for some commutative ring k , the ∞ -category of local systemson S with values in D ( k ) is equivalent to the ∞ -category D ( k [ t, t − ]), where k [ t, t − ] is thering of Laurent polynomials. In this section we show the existence of an equivalence of k -linear ∞ -categories Fun( S n , D ( k )) ≃ D ( k [ t n − ]) for n >
2, where k [ t n − ] the polynomial algebra withgenerator in degree | t n − | = n −
1. In Section 7.2, we will show that this description alsogeneralizes to D = RMod R for R a E ∞ -ring spectrum. We will end this section by an explicitdescription of the cotwist functor of the adjunction f ∗ ⊣ f ∗ . e begin with the following observation. Remark 5.2.
Let Z be a simplicial set. The ∞ -category Fun( Z, D ( k )) admits a symmetricmonoidal structure, such that the pullback functor h ∗ along h : Z → ∗ is a symmetric monoidalfunctor, see for example [Chr20, Section 3.3]. We can thus consider Fun( Z, D ( k )) as a leftmodule in P r L St over itself and the functor h ∗ as a morphism of algebra objects in P r L . Pullingback along h ∗ provides Fun( Z, D ( k )) with the structure of a left module over D ( k ) and thuswith the structure of a left-tensoring over D ( k ). This shows that Fun( Z, D ( k )) is a k -linear ∞ -category such that the functor h ∗ is k -linear.Consider now the simplicial set L consisting of a single vertex and a single non-degenerateedge. We use Remark 5.2 to lift Fun( L, D ( k )) to a k -linear ∞ -category. Denote by k [ t ] thepolynomial algebra with | t | = 0. Lemma 5.3.
Consider the morphism of simplicial sets g : L → ∗ and the associated pull-back functor g ∗ : D ( k ) → Fun( L, D ( k )) . There exists an equivalence of k -linear ∞ -categories Fun( L, D ( k )) ≃ D ( k [ t ]) such that the following diagram commutes. D ( k )Fun( L, D ( k )) D ( k [ t ]) g ∗ φ ∗ ≃ (32) Here φ ∗ denotes the pullback functor along the morphism of dg-algebras k [ t ] t −−−→ k .Proof. We observe that Fun( L, D ( k )) admits a compact generator X , given by the diagram k [ t ] · t −−→ k [ t ] in D ( k ). The homology of the k -linear endomorphism algebra End k ( X ) is concen-trated in degree 0 and a direct computation shows that it is equivalent to the k -vector space k [ t ].It follows that End k ( X ) is formal. The composition of morphisms induces the polynomial algebrastructure on k [ t ]. In total, this shows that End k ( X ) is quasi-isomorphic as a dg-algebra to k [ t ].The existence of the equivalence of k -linear ∞ -categories ǫ : Fun( L, D ( k )) ≃ D ( k [ t ]) thus followsfrom Lemma 2.5. The k -linear functors g ∗ , φ ∗ are fully determined by g ∗ ( k ) = ( k id −→ k ), re-spectively, φ ∗ ( k ), see [Lur17, Section 4.8.4]. Clearly, there exists an equivalence ǫg ∗ ( k ) ≃ φ ∗ ( k ),showing the commutativity of the diagram (32).For n ≥
1, we denote by i : ∗ → S n the inclusion functor of any vertex and by i ∗ :Fun( S n , D ( k )) → D ( k ) the associated pullback. We use Remark 5.2 to lift i ∗ and the functor f ∗ from (31) to k -linear functors. We further denote by g ! , f ! and i ! the left adjoints of g ∗ , f ∗ ,respectively, i ∗ . Lemma 5.4.
1. There exists a pushout diagram in
LinCat k as follows. Fun( L, D ( k )) D ( k ) D ( k ) Fun( S , D ( k )) g ! g ! p i ! i ! (33)
2. Let n ≥ . There exists a pushout diagram in LinCat k as follows. Fun( S n − , D ( k )) D ( k ) D ( k ) Fun( S n , D ( k )) f ! f ! p i ! i ! (34) roof. We begin by showing statement 2. Consider the following pushout diagram of spaces. S n − ∗∗ S nff p ii The above diagram is also pushout in Cat ∞ . Applying the limit preserving functor Fun(- , D ( k )) :Cat op ∞ → Cat ∞ maps this pushout diagram to the following pullback diagram in P r R .Fun( S n , D ( k )) D ( k ) D ( k ) Fun( S n − , D ( k )) i ∗ i ∗ y f ∗ f ∗ The left adjoint diagram is the diagram (34) and thus pushout in LinCat k .We now show statement 1. The geometric realization of L is equivalent to the topological1-sphere. There thus exists a morphism of simplicial sets L → S such that the limit functor g ∗ = lim : Fun( L, D ( k )) → D ( k ) restricts via the pullback functor i ∗ : Fun( S , D ( k )) → Fun( L, D ( k )) to the limit functor f ∗ . The left adjoint g ∗ : D ( k ) → Fun( L, D ( k )) thus factorsthrough Fun( S , D ( k )). It thus follows from the explicit model for limits in Cat ∞ that the rightadjoint diagram of diagram (33) is pullback in P r R . It follows that the diagram (33) is pushoutin LinCat k . Proposition 5.5.
Let n ≥ . There exists an equivalence of k -linear ∞ -categories Fun( S n , D ( k )) ≃ D ( k [ t n − ]) , (35) such that the following diagram in LinCat k commutes. Fun( S n , D ( k )) D ( k [ t n − ]) D ( k )Fun( S n , D ( k )) D ( k [ t n − ]) ≃ i ∗ Gf ∗ φ ∗ ≃ (36) Here G denoted the monadic functor and φ ∗ the pullback functor along the morphism of dg-algebras φ : k [ t n − ] t n − −−−−−→ k .Proof. We observe that the composition of the autoequivalence of dg-algebras k [ t ] t t − −−−−−−→ k [ t ] with the morphism of dg-algebras k [ t ] t −−−→ k [ t ] is given by k [ t ] t −−−→ k . It thereforefollows from Lemma 5.3, that for n = 2 the pushout square (34) is equivalent to the image under ψ of the following homotopy pushout diagram of dg-categories with a single object. k [ t ] k ( k [ t , t ] , d ) k [ t ] t t t p Above k [ t , t ] denotes the polynomial dg-algebra in two variables with generators in degrees0 and 1 and differential d determined by d ( t ) = t , d ( t ) = 0. It follows that there exists an quivalence of k -linear ∞ -categories Fun( S , D ( k )) ≃ D ( k [ t ]), making the the upper half of(36) commute. The k -linear functors φ ∗ : D ( k ) → D ( k [ t ]) and f ∗ : D ( k ) → Fun( S , D ( k )) aredetermined by φ ∗ ( k ) and f ∗ ( k ), respectively. The homology of the chain complex underlyingthe right k [ t ]-module φ ∗ ( k ) is concentrated in degree 0, given by k . Using that the upper half of(36) commutes and that i ∗ f ∗ ( k ) = k , it follows that f ∗ ( k ) is also mapped under the equivalenceFun( S , D ( k )) ≃ D ( k [ t ]) to a right k [ t ]-module with homology k . There exists but a uniqueright k [ t ]-module, up to quasi-isomorphism, with homology k . This right k [ t ]-module k isequivalent to the module determined by the morphism of dg-algebras k [ t ] t −−−→ k . It followsthat the lower half of (36) commutes for n = 2. For n >
3, one can argue analogously andby induction. The pushout square (34) is equivalent to the image under ψ of the followinghomotopy pushout diagram of dg-categories. k [ t n − ] k ( k [ t n − , t n − ] , d ) k [ t n − ] t n − t n − t n − p Here again k [ t n − , t n − ] denotes the polynomial dg-algebra in two variables with generators indegrees n − n − d ( t n − ) = t n − , d ( t n − ) = 0. We findthe desired equivalence of k -linear ∞ -categories Fun( S n , D ( k )) ≃ D ( k [ t n − ]) making the upperhalf of (36) commute. Showing that the lower half of (36) commutes is analogous to the case n = 2. Remark 5.6.
The composite functor ∗ i −→ S n f −→ ∗ is an equivalence of spaces. Denote the leftadjoints of i ∗ and f ∗ by i ! and f ! , respectively. The functor D ( k ) i ! −→ Fun( S n , D ( k )) f ! −→ D ( k )is thus equivalent to id D ( k ) . By sphericalness, there exists an equivalence f ∗ ≃ f ! [ − n ] andtherefore f ∗ i ! ( k ) ≃ k [ − n ]. Note that Proposition 5.5 shows that i ! ( k ) is equivalent to thecompact generator k [ t n − ] ∈ D ( k [ t n − ]) ≃ Fun( S n , D ( k )).We now describe the cotwist functor T Fun( S n , D ( k )) of the spherical adjunction f ∗ ⊣ f ∗ for n ≥
2. Consider the morphism of dg-algebras ϕ : k [ t n − ] → k [ t n − ] determined by ϕ ( t n − ) =( − n +1 t n − . Let ϕ ∗ : D ( k [ t n − ]) → D ( k [ t n − ]) be the pullback functor. Proposition 5.7.
Let n ≥ and let k be a commutative ring. There exists a commutativediagram in LinCat k as follows. Fun( S n , D ( k )) Fun( S n , D ( k )) D ( k [ t n − ]) D ( k [ t n − ]) T Fun(
Sn, D ( k )) ≃ (35) ≃ (35) ϕ ∗ [ − n ] Remark 5.8.
Note that only if n is odd there exists an equivalence of functors T Fun( S , D ( k )) ≃ [ − n ]. The functor T Fun( S , D ( k )) [ n ] is otherwise the involution reversing the sign of t n − . Proof of Proposition 5.7.
In the following, let ⊙ denote the multiplication in k [ t n − ]. We denoteby \ k [ t n − ] the k [ t n − ]-bimodule \ k [ t n − ] with• underlying chain complex k [ t n − ],• left action on a ∈ \ k [ t n − ] determined by t in − .a = ( − i ( n − t in − ⊙ a and right action on a ∈ \ k [ t n − ] determined by a.t in − = a ⊙ t in − .Note that \ k [ t n − ] ≃ φ ∗ ( k [ t n − ]). We can thus show the equivalent statement that the composi-tion of the twist functor T ′ Fun( S n , D ( k )) of the spherical adjunction f ! ⊣ f ∗ with the equivalence(35) is equivalent to - ⊗ \ k [ t n − ][ n ]. Using the commutativity of the lower part of diagram (36) inProposition 5.5, it suffices to show that the twist functor T of the spherical adjunction φ ! ⊣ φ ∗ is equivalent to - ⊗ \ k [ t n − ][ n ].Using Remark 5.6, it follows that φ ∗ φ ! ( k [ t n − ]) ≃ k ∈ D ( k [ t n − ]) with k the k [ t n − ]-moduledetermined by the morphism of dg-algebras φ . The k -linear functor φ ∗ φ ! is thus equivalent tothe functor - ⊗ k [ t n − ] k , for a k [ t n − ]-bimodule k . There is but a unique such bimodule, whichcarries the action t n − . .t n − ∈ k . A cofibrant replacement of the k [ t n − ]-bimodule k in dgMod( k [ t n ] ⊗ k k [ t n − ] op ) with the projective model structure is given the cone of themorphism of bimodules α : \ k [ t n − ][ n − → k [ t n − ] .t in − t in − To see that α indeed exists, note that by the definition of \ k [ t n − ] and the sign rule for the shiftof left modules, see Remark 2.9, the left action of k [ t n − ] on \ k [ t n − ][ n −
1] is determined by t n − . − ( n − n − t n − = t n − . We deduce that the twist functor T is equivalent to thefunctor given by tensoring with the homotopy pushout in the following diagram of cofibrant k [ t n − ]-bimodules. k [ t n − ] cone( α )0 \ k [ t n − ][ n ] p We have shown T ≃ - ⊗ k \ k [ t n − ][ n ], finishing the proof. V f ∗ and V f ∗ For n ≥
2, we consider the spherical adjunction f ∗ : D ( k ) ↔ Fun( S n , D ( k )) : f ∗ of Propo-sition 5.1.9 The goal of this section is to prove Propositions 5.9 and 5.12 below, describingthe parametrized perverse schober on a 1-gon, 2-gon and 3-gon obtained from the sphericaladjunction f ∗ ⊣ f ∗ . Proposition 5.9.
Let k be a commutative ring and n ≥ . Let D = k and let D be the freelygenerated dg-category with two objects y, z depicted as follows, y z bb ∗ with morphisms in degrees | b | = 0 and | b ∗ | = n − and vanishing differentials. Let further D be the freely generated dg-category with three objects x, y, z depicted as follows, yx z a ∗ ba c ∗ b ∗ c with morphisms in degrees | b | = | c | = 0 , | a | = n − , | b ∗ | = | c ∗ | = n − , | a ∗ | = 1 and differentialdetermined by d ( a ) = d ( b ) = d ( c ) = 0 , ( a ∗ ) = c ∗ b ∗ , d ( b ∗ ) = a ∗ c ∗ , d ( c ∗ ) = b ∗ a ∗ . There exist equivalences of ∞ -categories N f ∗ ≃ ψ ( k [ t n − ]) , (37) V f ∗ ≃ ψ ( D ) , (38) V f ∗ ≃ ψ ( D ) , (39) V f ∗ ≃ ψ ( D ) , (40) Remark 5.10.
Proposition 5.9 shows that dimension n = 2 is distinguished. Only in dimension n = 2, are the morphisms a, b, c in D all in degree 0.In the gluing construction for the Ginzburg algebra we will only need to consider the sphericaladjunction f ∗ ⊣ f ∗ for n = 2. We thus state and prove all further results in this section underthe assumption that n = 2. Notation 5.11.
1. Denote by i +1 = i − : k [ t n − ] → D = k the dg-functor determined by mapping l to 0.2. For τ = + , − , denote by i τ , i τ : k [ t n − ] → D the dg-functors determined by mapping ∗ to z and y , respectively and l to τ bb ∗ and − τ b ∗ b , respectively.3. For τ = + , − , denote by i τ , i τ , i τ : k [ t n − ] → D the dg-functors determined by mapping ∗ to x, z and y , respectively, and l to τ ( cc ∗ − a ∗ a ), τ ( bb ∗ − c ∗ c ) and τ ( aa ∗ − b ∗ b ), respectively. Proposition 5.12.
Let n = 2 .1. Under the equivalences (37) and (38) , the functor ς : N f ∗ → V f ∗ is equivalent to theimage under ψ of the dg-functor i +1 : k [ t n − ] → D .2. Under the equivalences (37) and (39) , the functors ς , ς : N f ∗ → V f ∗ are equivalent to theimage under ψ of the dg-functors i +1 , i − : k [ t n − ] → D , respectively.3. Under the equivalences (37) and (40) , the functors ς , ς , ς : N f ∗ → V f ∗ are equivalent tothe image under ψ of the dg-functors i +1 , i − , i +3 : k [ t n − ] → D , respectively. Lemma 5.13.
Let n ≥ .1. There exist an equivalence of ∞ -categories V f ∗ ≃ D ( C ) , (41) where C is the full dg-subcategory of dgMod( A ) spanned by p i A for i = 1 , , see alsoNotation 2.41, where A = (cid:18) k k [ − n ]0 k [ t n − ] (cid:19) is the upper triangular dg-algebra.2. There exist an equivalence of ∞ -categories V f ∗ ≃ D ( C ) , where C is the full dg-subcategory of dgMod( A ) spanned by p i A for i = 1 , , , where A = k k [ − n ] 00 k [ t n − ] k [ t n − ]0 0 k [ t n − ] is the upper triangular dg-algebra and where k [ t n − ] denotes the canonical k [ t n − ] -bimodule. roof. This follows from Proposition 2.40 and the characterization of f ∗ in Proposition 5.5 aswell as Remark 5.6. Proof of Proposition 5.9.
We have constructed the equivalence (37) in Proposition 5.5. Theexistence of the equivalence (38) is immediate.We now prove the existence of the equivalence (39). Consider the compact generators w = e A [ − n ], y = e A , of D ( A ). Let z = cof( w e −→ y ) ∈ D ( A ) with e being the edgecorresponding to 1 ∈ k [ n ] ≃ Hom dgMod( A ) ( w, y ). The A -module z is cofibrant as the total-ization of a cofibrant resolution. Note that y ⊕ z is a compact generator of D ( A ). We thusobtain an equivalence of k -linear ∞ -categories D ( C ) ≃ RMod
End k ( y ⊕ z ) . The right A -module z is equivalent to the module obtained from the following pushout and homotopy pushout indgMod( A ). w cone( id w ) y z p We note also that z is also equivalent in dgMod( A ) to the module obtained from the followingpullback and homotopy pullback in dgMod( A ). z w [1]cone( id y ) y [1] y Using the universal properties of the above pullback and pushout, we can compute the followingHom-complexes in dgMod A . We findHom( z, z ) ≃ Hom( w [1] , w [1]) | {z } ≃ k ⊕ Hom( w [1] , y ) | {z } ≃ k [ − ⊕ Hom( y, y ) | {z } ≃ k [ t n − ] , (42)with the differential on the right side determined by1 ∈ k ≃ Hom( w [1] , w [1]) ∈ k [ − ≃ Hom( w [1] , y ) ,t in − ∈ k [ t n − ] ≃ Hom( y, y ) ( i = 0 , , ∈ k [ − ≃ Hom( w [1] , y ) . The splitting of the right side in (42) holds, of course, only on the level of graded k -modules.We further compute Hom( z, y ) ≃ Hom( w [1] , y ) ⊕ Hom( y, y )with the differential on the right side determined by t in − ∈ k [ t n − ] ≃ Hom( y, y ) ( i = 0 , , ∈ k [ − ≃ Hom( w [1] , y ) , and Hom( y, z ) ≃ Hom( y, y ) ≃ k [ t n − ] . The above Hom-complexes together with their composition rules determines the full dg-subcategoryof dgMod( A ) spanned by y, z , which we thus find to be quasi-equivalent to D . Using Proposi-tion 2.22, it follows that there exists an equivalence of dg-algebras End dg ( y ⊕ z ) ≃ End k ( y ⊕ z ).In total we thus find equivalences of ∞ -categories ψ ( D ) ≃ D (End dg ( y ⊕ z )) ≃ D (End k ( y ⊕ z )) ≃ ψ ( C ) (41) ≃ V f ∗ . (43) o later match some signs, we additionally compose the equivalence (43) with the equivalence ψ ( χ ) : ψ ( D ) ≃ ψ ( D ), where χ : D → D is the dg-functor satisfying χ ( y ) = y, χ ( z ) = z, χ ( b ) = b and χ ( b ∗ ) = − b ∗ . We choose the equivalence (39) to be the composite of ψ ( χ ) and(43).We proceed analogously for the equivalence (40). Consider the compact generators v = p A [ − n ] , y = p A , z = p A of D ( A ). Let w = cof( v e −→ y ) ∈ D ( A ) with e corresponding to 1 ∈ k [ n ] ≃ Hom dgMod( A ) ( v, y ).We again identify w with the cone of v e −→ y in dgMod( A ). We compute the following Hom-complexes in dgMod A . We findHom( w, w ) ≃ Hom( v [1] , v [1]) ⊕ Hom( v [1] , y ) ⊕ Hom( y, y ) , with the differential on the right determined by1 ∈ k ≃ Hom( v [1] , v [1]) ∈ k [ − ≃ Hom( v [1] , y ) ,t in − ∈ k [ t n − ] ≃ Hom( y, y ) ( i = 0 , , ∈ k [ − ≃ Hom( v [1] , y ) . We further find Hom( w, y ) ≃ Hom( v [1] , y ) ⊕ Hom( y, y )with the differential on the right side determined by t in − ∈ k [ t n − ] ≃ Hom( y, y ) ( i = 0 , , ∈ k [ − ≃ Hom( v [1] , y ) , and Hom( y, w ) ≃ Hom( y, y ) ≃ k [ t n − ] . Lastly, we compute Hom( w, z ) ≃ Hom( y, z ) ≃ k [ t n − ]and Hom( z, w ) ≃ . We denote the full dg-subcategory of dgMod( A ) spanned by w, y, z by ˆ C and by y ∐ z thedg-category with two objects y, z equivalent to the coproduct of the dg-categories k and k . Wefurther denote by D ( z, y ) the dg-category with two objects y, z andHom D ( z,y ) ( y, y ) = Hom D ( z,y ) ( z, z ) = k , Hom D ( z,y ) ( z, y ) = cone(id k [ n − ) , Hom D ( z,y ) ( y, z ) = 0 , and the obvious composition rules. Consider the pushout of dg-categories. y ∐ z D ( z, y )ˆ C ˆ C ′ α p β The above diagram is cofibrant, the pushout is also a homotopy pushout and α is a quasi-equivalence. It thus follows that the dg-functor β : ˆ C → ˆ C ′ is a cofibration and quasi-equivalence. We label the elements of ˆ C ′ by w ′ , y ′ , z ′ such β ( w ) = w ′ , β ( y ) = y ′ , β ( z ) = z ′ . e can describe the categories ˆ C (on the left) and ˆ C ′ (on the right) as generated by the follow-ing morphisms, labelled by their degrees. y y ′ w z w ′ z ′ n − n − n − n − n − n − The above descriptions are not completely accurate with respect to the morphisms w → y and w ′ → y ′ , the equivalences Hom( w, y ) ≃ k [ t n − ][1] and Hom( w ′ , y ′ ) ≃ k [ t n − ][1] only hold onhomology. However, up to quasi-equivalence of dg-categories, the above descriptions hold. Weleave this identification in the following implicit and refer the above depicted morphisms of ˆ C ′ as the generating morphisms of ˆ C ′ .We denote by x ′ the cone of the generating morphism w ′ → z ′ in dgMod( ˆ C ′ ). We denoteby B ′ the full dg-subcategory of dgMod( ˆ C ′ ) spanned by x ′ , y ′ , z ′ . We now describe the Homcomplexes in B ′ . Leaving the differential implicit, we find the following equivalences.Hom( x ′ , x ′ ) ≃ Hom( w ′ [1] , w ′ [1]) ⊕ Hom( w ′ [1] , z ′ ) ⊕ Hom( z ′ , w ′ [1]) ⊕ Hom( z ′ , z ′ ) (44)Hom( x ′ , y ′ ) ≃ Hom( w ′ [1] , y ′ ) ⊕ Hom( z ′ , y ′ ) (45)Hom( y ′ , x ′ ) ≃ Hom( y ′ , w ′ [1]) ⊕ Hom( y ′ , z ′ ) (46)Hom( x ′ , z ′ ) ≃ Hom( w ′ [1] , z ′ ) ⊕ Hom( z ′ , z ′ ) (47)Hom( z ′ , x ′ ) ≃ Hom( z ′ , w ′ [1]) ⊕ Hom( z ′ , z ′ ) (48)We identify a set M of generating morphisms of B ′ . We let M consist of• the morphism y ′ → z ′ in B ′ of degree 0 corresponding to the composite of the generatingmorphisms y ′ → w ′ → z ′ of ˆ C ′ and the morphism z ′ → y ′ in B ′ of degree n − z ′ → y ′ of ˆ C ′ .• the morphism x ′ → y ′ in B ′ of degree n − w ′ → y ′ of ˆ C ′ of degree n − y ′ → x ′ in B ′ of degree 1 correspondingto the generating morphism y ′ → w ′ of ˆ C ′ of degree 0.• the morphism x ′ → z ′ in B ′ of degree − w ′ → z ′ of ˆ C ′ of degree 0, the morphism x ′ → z ′ in B ′ of degree 0 corresponding to id z ′ ,the morphism x ′ → z ′ in B ′ of degree n − z ′ → z ′ of ˆ C ′ of degree n − z ′ → x ′ in B ′ of degree 0, correspondingto id z ′ .To show that the set M really generates all morphisms in B ′ , it suffices to show that M generatesa) any morphism in B ′ corresponding under one of the equivalences (44)-(48) to a generatingmorphism of ˆ C ′ in any of the summands,b) as well as the morphism in B ′ obtained from the generating morphism z ′ → z ′ of degree n − C ′ and the two morphisms in B ′ obtained from the generating morphisms z ′ → y ′ of degrees n − n − C ′ .Statement a ) can be shown for each the morphisms complexes in (44)-(48) separately. Weexemplify the argument for the generating morphisms in (44). The identity on x ′ in B ′ isthe morphism ( id w ′ [1] , id z ′ ) ∈ Hom( w ′ [1] , w ′ [1]) ⊕ Hom( z ′ , z ′ ) ⊂ Hom( x ′ , x ′ ). The morphism id z ′ ∈ Hom( z ′ , z ′ ) ⊂ Hom( x ′ , x ′ ) is obtained as the composite of morphisms x ′ id z ′ −−→ z ′ id z ′ −−→ x ′ in M . The morphism x ′ → x ′ in B ′ given by id w ′ [1] = id x ′ − id z ′ ∈ Hom( x ′ , x ′ ) is thus alsogenerated by M . The last remaining morphism in B ′ corresponds to a generating morphism inˆ C ′ given by h : w ′ [1] → z ′ of degree −
1, which is obtained as a morphism in B ′ as the composite f two morphisms x ′ → z ′ id z ′ −−→ x ′ in M of degrees − b ). The degree n − z ′ → y ′ in B ′ consideredin b ) is contained in M . We denote by h the morphism z ′ → z ′ in B ′ of degree n − b ) and by h the morphism z ′ → y ′ of degree n − B ′ considered in b ). The morphism h can be obtained as the composite of the morphism id z ′ ∈ Hom( z ′ , z ′ ) ⊂ Hom( z ′ , x ′ ) with themorphism t n − ∈ k [ t n − ] ≃ Hom( z ′ , z ′ ) ⊂ Hom( x ′ , z ′ ). The morphism h can be obtained asthe composite of the morphism id z ′ ∈ Hom( z ′ , z ′ ) ⊂ Hom( z ′ , x ′ ) in B ′ with the morphism inHom( z ′ , y ′ ) ⊂ Hom( x ′ , y ′ ) corresponding to the degree n − z ′ → y ′ inˆ C ′ . We now describe the differentials of the morphisms in M . The set M consists of eightmorphisms which are determined by their source, target and degree. We hence refer to themorphisms in M in the following by specifying that data. A direct computation shows that thedifferentials are as follows. d ( y ′ −−→ z ′ ) = 0 d ( z ′ n − −−−→ y ′ ) = ( x ′ n − −−−→ y ′ ) ◦ ( z ′ −→ x ′ ) d ( x ′ n − −−−→ y ′ ) = 0 d ( y ′ −−→ x ′ ) = ( z ′ −→ x ′ ) ◦ ( y ′ −→ z ′ ) d ( z ′ −−→ x ′ ) = 0 d ( x ′ − −−→ z ′ ) = 0 d ( x ′ −−→ z ′ ) = ( x ′ − −−→ z ′ ) d ( x ′ n − −−−→ z ′ ) = ( y ′ −→ z ′ ) ◦ ( x ′ n − −−−→ y ′ )It thus follows that there exists a quasi-equivalence D → B ′ , mapping x, y, z to x ′ , y ′ , z ′ .Proposition 2.22 shows there exists a quasi-isomorphism of dg-algebras End dg ( x ′ ⊕ y ′ ⊕ z ′ ) ≃ End k ( x ′ ⊕ y ′ ⊕ z ′ ) . In total we obtain equivalences of ∞ -categories ψ ( D ) ≃ ψ ( B ′ ) ≃ D (End dg ( x ′ ⊕ y ′ ⊕ z ′ )) ≃ D (End k ( x ′ ⊕ y ′ ⊕ z ′ )) ≃ D ( ˆ C ′ ) ≃ D ( ˆ C ) ≃ D ( C ) ≃ V f ∗ , which we choose as the equivalence (40). Proof of Proposition 5.12.
Statement 1 follows from Proposition 5.5.We now show statement 3 and deduce statement 2 below. In the proof of Proposition 5.9we introduced the dg-category ˆ C with objects w, y, z . We denote x = cof( w e −→ z ) ∈ D ( ˆ C ),where e is the morphism corresponding to 1 ∈ k [ t ] ≃ Hom dgMod( ˆ C ) ( w, z ). Note that we canidentify x in dgMod( ˆ C ) with the mapping cone of e . We denote by B the full dg-subcategoryof dgMod( A ) spanned by x, y, z . We begin by describing the Hom complexes in B . We findHom( x, x ) ≃ Hom( w [1] , w [1]) ⊕ Hom( w [1] , z ) ⊕ Hom( z, z ) , with the differential on the right side determined by t i ∈ k [ t ] ≃ Hom( w [1] , w [1]) t i ∈ k [ t ][ − ≃ Hom( w [1] , z ) ,t i ∈ k [ t ] ≃ Hom( z, z ) ( − i t i ∈ k [ t ][ − ≃ Hom( w [1] , z ) , and Hom( x, y ) ≃ Hom( w [1] , y ) ≃ k [ t ] , om( y, x ) ≃ Hom( y, w [1]) ⊕ Hom( y, z ) , where the differential of the latter is determined by t i ∈ k [ t ][1] ≃ Hom( y, w [1]) t i ∈ k [ t ] ≃ Hom( y, z ) . Note that Hom( y, x ) is acyclic. We further computeHom( x, z ) ≃ Hom( w [1] , z ) ⊕ Hom( z, z ) , with the differential determined by t i ∈ k [ t ] ≃ Hom( z, z ) ( − i t i ∈ k [ t ][ − ≃ Hom( w [1] , z ) . Hom( x, z ) is thus also acyclic. Lastly, we obtainHom( z, x ) ≃ Hom( z, z ) ≃ k [ t ] . We can thus describe the dg-category B as generated by morphisms, yx z
010 10 labelled by their degrees, subject to the relation that the morphism z → x → y of degree 0vanishes and leaving the differentials implicit. The functors ς , ς [1] : Fun( S , D ( k )) → V f ∗ arethe inclusions of the third, respectively, second component of the semiorthogonal decompositionof V f ∗ . Using Proposition 2.40, we obtain that the functor ς is modelled by the dg-functor σ : k [ t ] → C determined by mapping the unique object ∗ of k [ t ] to z and t to the endomorphismof degree 1 of z given by t ∈ k [ t ] ≃ Hom C ( z, z ). Let further σ ′ : k [ t ] → B be the dg-functordetermined by mapping ∗ to z and t to the composite of the generating morphisms z −→ x −→ z and σ ′′ : k [ t ] → D determined by mapping ∗ to z ′ and t to the c ∗ c − bb ∗ . In total, there isthe following commutative diagram of dg-categories k [ t ] C perf3 B perf ( D ) perf( σ ) perf ( σ ′ ) perf ( σ ′′ ) perf where the horizontal edges are all quasi-equivalences, showing that ς is modelled by σ ′′ . Ananalogous argument shows that ς is modelled by the dg-functor σ : k [ t ] → D determined bymapping ∗ to y ′ and t to aa ∗ − b ∗ b . Using the sequence of adjunctions ς ⊣ ̺ ⊣ ς ⊣ ̺ ⊣ ς and the rotational symmetry of D it follows that ς is modelled by the dg-functor k [ t ] → D determined by mapping ∗ to x ′ and t to cc ∗ − a ∗ a .We now show statement 2. To clarify notation, we denote the right adjoints of the categorifiedrestrictions maps ̺ , ̺ : Fun( S , D ( k )) → V f ∗ by ˜ ς and ˜ ς , respectively, instead of ς and ς .Note that the following diagrams of ∞ -categories commute.Fun( S n , D ( k )) V f ∗ V f ∗ ς ˜ ς π , Fun( S n , D ( k )) V f ∗ V f ∗ ς ˜ ς [ − π , (49) he functor ς is modelled by the dg-functor i +3 : k [ t ] → D and the functor ς is modelledby the dg-functor i +1 : k [ t ] → D . The functor π , is modelled by the dg-functor π x,y : D → dgMod( D ), mapping the objects x, y, z to y [ − , z,
0, respectively, and mapping a to − b ∗ (the sign arises from (43)) and a ∗ to b . It follows that ˜ ς is modelled by the dg-functor k [ t ] → dgMod( D ), determined by mapping the unique object ∗ of k [ t ] to z and t to bb ∗ andthat ˜ ς [ −
1] is modelled by the dg-functor k [ t ] → D determined by mapping ∗ to y [ −
1] and t to − b ∗ b . Using the equivalence of functors ˜ ς ≃ T − V f ∗ ◦ ˜ ς , we can deduce from Lemma 5.14that ˜ ς is modelled by the dg-functor k [ t ] → T V f ∗ determined by mapping ∗ to z and t to b ∗ b .This concludes the argument.We end this section with describing a dg-model for the paracyclic autoequivalences T V f ∗ and T V f ∗ of V f ∗ , respectively, V f ∗ as defined in Proposition 3.11. Lemma 5.14.
Let n = 2 .1. Let T V f ∗ be the paracyclic autoequivalence of V f ∗ of Proposition 3.11. Denote by α : ψ (dgMod( D )) ≃ T V f ∗ the equivalence of ∞ -categories obtained from composing the equiv-alence ψ (cid:0) D −−−−→ dgMod( D ) (cid:1) with the equivalence (39) . The Morita equivalence ofdg-categories T D : D → dgMod( D ) given by y, z z [ − , y ,b b ∗ ,b ∗ b , makes the following diagram of ∞ -categories commute. V f ∗ V f ∗ ψ ( D ) ψ (dgMod( D )) T V f ∗ (39) αψ ( T D )
2. Let T V f ∗ be the paracyclic autoequivalence of V f ∗ of Proposition 3.11. The equivalence ofdg-categories T D : D → D given by x, y, z y, z, x ,a, b, c b, c, a ,a ∗ , b ∗ , c ∗
7→ − b ∗ , − c ∗ , − a ∗ , makes the following diagram of ∞ -categories commute. V f ∗ V f ∗ ψ ( D ) ψ ( D ) T V f ∗ (40) (40) ψ ( T D ) Proof.
We only prove the second statement, the first statement can be proven completely anal-ogously, using the models for ς and ς [ −
1] from the proof of Proposition 5.12. Instead of thetwist functor T V f ∗ of the spherical adjunction F ′ ⊣ G ′ we characterize its inverse T ′ V f ∗ given by he cotwist functor of the spherical adjunction E ′ ⊣ F ′ . There exists a commutative diagram of ∞ -categories, V f ∗ V f ∗ ψ ( D ) ψ ( D ) T ′ V f ∗ (40) (40) ψ ( T ′ ) where T ′ is the cotwist functor of the spherical adjunction( ψ ( i +1 ) , ψ ( i − ) , ψ ( i +3 )) : D ( k [ t n − ]) × ←→ D ( D ) : (RHom D ( x, -) , RHom D ( z, -) , RHom D ( y, -)) . We consider x, y, z ∈ D as objects of D ( D ) and a, a ∗ , b, b ∗ , c, c ∗ as the corresponding morphismsin D ( D ) in the following. The cotwist functor T ′ is k -linear and satisfies T ( x ) = z, T ( y ) = x, T ( z ) = y . Combining statement 3 of Proposition 5.12 and Proposition 3.11 we further obtain T ′ ( b ∗ b − aa ∗ ) = cc ∗ − a ∗ a , (50) T ′ ( cc ∗ − a ∗ a ) = c ∗ c − bb ∗ , (51) T ′ ( c ∗ c − bb ∗ ) = aa ∗ − b ∗ b . (52)By definition, the cotwist functor T ′ maps the morphism a : x → y to the fiber in the verticaldirection of the following diagram in D ( D ), x ⊕ z y ⊕ xx y ( id x ,c ) M ( id y ,a ) a where M = (cid:18) a c (cid:19) . This fiber is given by c : z → x . Similar computations show that T ′ ( b ) = a and T ′ ( c ) = b . In combination with (50)-(52), we obtain T ′ ( a ∗ ) = − c ∗ , T ′ ( b ∗ ) = − a ∗ and T ′ ( c ∗ ) = − b ∗ . The statement now follows from the observation that ψ ( T ′ ) ≃ ψ ( T D ) − . We fix a commutative ring k . We begin by associating a relative Ginzburg algebra G B T to anymarked surface S with an ideal triangulation T and subset B of the set of boundary edges of T . The set B determines which parts of the boundary are not incorporated in G B T . If thereare no self-folded triangles and the boundary of S is empty, or B the set of all boundary edges, G B T ≃ G ( Q ◦ T , W ′ T ) recovers a Ginzburg algebra associated to T . Definition 6.1.
Let T be an ideal triangulation of a surface and let B be a subset of the set ofboundary edges of T . We denote by Q B T the quiver determined as follows.i) The set of vertices ( Q B T ) is the set of edges of T not lying in B . We call these edgesadmissible.ii) For each not self-folded ideal triangle f containing three admissible edges e f , e ′ f , e ′′ f , orderedin the clockwise ordering, there are arrows e f a f, −−→ e ′ f , e ′ f a f, −−→ e ′′ f , e ′′ f a f, −−→ e f of degree 0and arrows e ′ f a ∗ f, −−→ e f , e ′′ f a ∗ f, −−→ e ′ f , e f a ∗ f, −−→ e ′′ f of degree 1 in Q B T . ii) For each not self-folded ideal triangle f containing exactly two admissible edges e f , e ′ f ,ordered such that e ′ f follows e f in the clockwise ordering, there are arrows e f a f, −−→ e ′ f and e ′ f a ∗ f, −−→ e f of degree 0 and 1, respectively, in Q B T .iv) For each self-folded triangle f containing two admissible edges, with e f and e ′ f the edgescorresponding to the loop, respectively, the folded side in T , there are arrows e f a f, −−→ e ′ f , e ′ f a f, −−→ e f of degree 0 and e ′ f a ∗ f, −−→ e f , e f a ∗ f, −−→ e ′ f of degree 1 in Q B T . There are furtherloops a f, and a ∗ f, at e f of degrees 0, respectively, 1.v) For each self-folded triangle f containing exactly one admissible edge e f , given by theself-folded edge of T , there are loops a f, and a ∗ f, at e f of degrees 0, respectively, 1.vi) For each interior edge e , there is a loop l e, of degree 2 at the corresponding vertex in Q B T .We define a dg-algebra G B T = ( kQ B T , d ) with underlying graded algebra given by the path algebraof the quiver Q B T and differential determined by its nonzero actions on the following generators.• l e, P ( f,j ) p e [ a f,j , a ∗ f,j ] p e , where the sum runs over all ideal triangles f and j ∈ { , , } if f is as in ii ) or iii ) and j = 1 if f is as in iv ) or v ) and p e denotes the lazy path at thevertex corresponding to the interior edge e .• a ∗ f,j a f,j +2 ◦ a f,j +1 for all ideal triangles f as in ii ) or iii ) and where j, j + 1 , j + 2 areconsidered modulo 3.We refer to G B T as a relative Ginzburg algebra. If B = ∅ , we denote G T := G B T and Q T := Q B T .If B is the set of all boundary edges, we denote Q ◦ T := Q B T .Let again T be an ideal triangulation of a marked surface and B a subset of the set ofboundary edges. Consider the dual ribbon graph with singularities (Γ T , V T ), see Definition 4.10.We denote by B c = (Γ T ) \ B ⊂ (Γ T ) the complement of B . We want to define a (Γ T , V T )-parametrized perverse schober F T with the property that the spherical adjunction at each vertexis given by f ∗ : D ( k ) ↔ Fun( S , D ( k )) : f ∗ and such that the ∞ -category H B c (Γ T , F T ) of globalsections with support on B c is equivalent to the derived ∞ -category of the relative Ginzburgalgebra G B T . As we show in the proof of Theorem 6.2, we can achieve this by making for eachvertex of Γ T a choice of total order compatible with the cyclic order of the halfedges at thevertex and then defining F T via a gluing of the local models of a parametrized perverse schoberdefined in Definition 3.4. To match the signs in the differential of the Ginzburg algebra weneed to carefully choose the gluing diagrams. These choices will involve the autoequivalence ϕ ∗ of D ( k [ t ]) (35) ≃ Fun( S , D ( k )), where ϕ : k [ t ] t t −−−−−→ k [ t ] is the morphism of dg-algebrasdetermined by reversing the sign of the generator t . This construction depends a priori on thechoices of total orders. We discuss in Section 7.1 an interpretation of all involved choices interms of spin structures on the open surface S \ ( M ∩ S ◦ ) and why the resulting parametrizedperverse schober is up to equivalence independent of the choices made. Theorem 6.2.
Let T be an ideal triangulation of an oriented marked surface and B a subsetof the set of boundary edges of T . Consider the dual ribbon graph with singularities (Γ T , V T ) .Denote by B c ⊂ (Γ T ) the subset of edges whose corresponding edge of T is not contained in B . Then there exists a (Γ T , V T ) -parametrized perverse schober F T and an equivalence of ∞ -categories H B c (Γ T , F T ) ≃ D ( G B T ) . Restricting Theorem 6.2 to B c = (Γ T ) ◦ , we obtain the following. Corollary 6.3.
Let T be an ideal triangulation of an oriented marked surface and consider theassociated quiver Q ◦ T with -cyclic potential W ′ T , containing a -cycle for each interior face of T , see Section 1.1. There exists an equivalence of ∞ -categories H Γ ◦ T (Γ , F T ) ≃ D ( G ( Q ◦ T , W ′ T )) . roof of Theorem 6.2. We proof the statement by an induction on the number of ideal trianglesof T and a comparison of the computation of H B c (Γ T , F T ) with an explicit computation ofhomotopy colimits in dgCat k with the Morita model structure. Before that, we need to discussan equivalent description of the ∞ -category H B c (Γ T , F T ).Let (Γ ′ T , V T ) be the ribbon graph with singularities obtained from (Γ T , V T ) as in Lemma 4.19.Let further (Γ B T , V T ) be the ribbon graphs with singularities obtained from (Γ T , V T ) by removingall edges in B . Given a (Γ T , V T )-parametrized perverse schober F , we can first extend it to a(Γ ′ T , V T )-parametrized perverse schober F ′ T as in Lemma 4.19. Consider the contraction of ribbongraphs with singularities c : (Γ ′ T , V T ) → (Γ B T , V T ) given by contracting all edges connecting toa non-singular vertex (these are indexed by B ). Using Proposition 4.26, we obtain a (Γ B T , V T )-parametrized perverse schober F B T = c ∗ ( F ′ T ) satisfying H B c (Γ T , F T ) ≃ H (Γ ′ T , F ′ T ) ≃ H (Γ B T , F B T ).We define a dg-category C B T with objects the vertices of Q B T and morphisms generated bythe arrows of Q B T and together with two added endomorphisms l ′ e, , l ′ e, of degrees 1 , e . The differential d of the mapping complexes actsnontrivially only on the generators l ′ e, , l e, and a ∗ p,i , on the latter two it acts as the differentialof G B T and acts on l ′ e, as l ′ e, l ′ e, − X ( f,j ) p e [ a f,j , a ∗ f,j ] p e , where we use the same notation for the second summand as in the definition of the differentialof l e, in Definition 6.1. The dg-category C B T is Morita equivalent to G B T . For each vertex e corresponding to an admissible boundary edge, we denote by i + e , i − e : k [ t ] → C B T the dg-functorsdetermined by mapping the unique vertex ∗ of k [ t ] to e ∈ C B T and the generator t to l ′ e, and − l ′ e, , respectively.We deduce the theorem from the following statements, which we prove by an induction onthe number of ideal triangles of T . Let T be an ideal triangulation of a marked surface with dual ribbon graph with singularities (Γ T , V T ) and let B be a subset of the set of boundary edges. There exists a (Γ T , V T ) -parametrizedperverse schober F T with the following properties.1. Consider the (Γ B T , V T ) -parametrized perverse schober F B T obtained from F T as above. Thereexists an equivalence of ∞ -categories H (Γ B T , F B T ) ≃ ψ ( C B T ) , where ψ is the functor (11) .2. For each admissible boundary edge e , consider the restriction functor res e to the corre-sponding external edge of Γ B T . The left adjoint of the functor ψ ( C B T ) ≃ H (Γ B T , F B T ) res e −−→ Fun( S , D ( k )) ≃ D ( k [ t ]) is equivalent to ψ ( i + e ) or ψ ( i − e ) . Base case of the induction
Let T be the ideal triangulation consisting of a single not self-folded ideal triangle. Wechoose a total order on the edges of T compatible with the cyclic order induced by the counter-clockwise orientation. This provided us with an equivalence of posets Exit(Γ T ) ≃ C . Wecan thus define F T as the parametrized perverse schober G ( f ∗ ) described in Definition 3.4corresponding to the spherical adjunction f ∗ ⊣ f ∗ . The parametrized perverse schober F B T isthen given by G m ( f ∗ ) with m = 3 − | B | . Note that H (Γ T , F B T ) ≃ V mf ∗ . If m = 1 ,
3, the twodesired statements then follow from Propositions 5.9 and 5.12. Denote by T the equivalence un( S , D ( k )) ≃ D ( k [ t ]) ϕ ∗ −−→ D ( k [ t ]) ≃ Fun( S , D ( k )). If m = 2, the left dual of F B T is givenby the diagram Fun( S , D ( k )) V f ∗ Fun( S , D ( k )) ς ◦ T [ − ς Using the twist functor T V f ∗ , we find a natural equivalence between the above diagram and thefollowing diagram. Fun( S , D ( k )) V f ∗ Fun( S , D ( k )) ς ς The two statements now follow Propositions 5.9 and 5.12.Let now be T the triangulation consisting of a single self-folded triangle. We define F T using Notation 4.15 as follows. f ∗ ( ̺ ,̺ ) ̺ (53)The limit H (Γ T , F T ) is equivalent to the colimit of the left dual colim P r L D L F T , which is equivalentto the following coequalizer in P r L .Fun( S , D ( k )) V f ∗ ς ς ◦ T (54)The colimit of (54) is in turn equivalent to the pushout of the following span in P r L .Fun( S , D ( k )) × V f ∗ Fun( S , D ( k )) id ⊕ id ( ς ,ς ◦ T ) (55)Let T ′ be the ideal triangulation consisting of a single not self-folded triangle and denote by e , e any two boundary edges of T ′ . Diagram (55) is equivalent to the image under ψ of thefollowing diagram in dgCat k . k [ t ] ∐ k [ t ] C ∅ T ′ k [ t ] id ∐ id ( i + e ,i − e ) (56)The morphism ( i + e , i − e ) in (56) is a cofibration, we can thus compute the colimit of (56) as the(ordinary) colimit in the 1-category of k -linear dg-categories. This colimit is quasi-equivalent to C ∅ T , showing the desired equivalence H (Γ T , F T ) ≃ ψ ( C T ). The left adjoint of the restrictionfunctor D ( G B, cf T ) → D ( k [ t ]) to the external edge e factors as D ( k [ t ]) ψ ( i − e ) −−−−→ ψ ( C ∅ T ′ ) −→ ψ ( C ∅ T ) , where e is the corresponding edge of T ′ , and is thus equivalent to ψ ( i − e ). If T is the triangulationconsisting of a single self-folded triangle and B is maximal, i.e. contains the boundary edge, then F B T is given in the graphical notation as follows. f ∗ ( ̺ ,̺ ) gain twisting by T − V f ∗ , it follows that the right dual of F B T is naturally equivalent to the imageunder ψ of k [ t ] D i − i +1 A simple computation, similar to the one above, then shows that H (Γ T , F B T ) ≃ ψ ( C B T ). Thiscompletes the base case of the induction where the ideal triangulation consists of a single idealtriangle. Induction step
Assume now that the statement has been shown for all ideal triangulations with at most n ideal triangles. Let T n +1 be an ideal triangulation with n +1 ideal triangles and B a subset of theboundary edges. We choose any subtriangulation T n consisting of n connected ideal trianglesof T n +1 . The complement of T n in T n +1 consists of a single ideal triangle and is denoted by T .Denote by S the set of edges along which T and T n are glued in T n +1 and s = | S | . Considerthe subsets of boundary edges B = { e ∈ B | e lies in T } and B n = { e ∈ B | e lies in T n } of T and T n , respectively. To define the (Γ T n +1 , V T n +1 )-parametrized perverse schober F T n +1 wewish to glue F T and F T n in the sense of Lemma 4.17. To obtain the correct signs in thedifferentials of the relative Ginzburg algebra G T , we modify the diagram F T n : Exit(Γ T n ) → Stby postcomposing the functor F T n ( v (cid:1) e ), for each edge v → e in Exit(Γ T n ) where e is in S ,with an equivalence T e of Fun( S , D ( k )). We now describe T e . Let e be an edge in S . Wedenote by i e the left adjoint of D ( G T ) ≃ H (Γ T , F T ) res e −−→ Fun( S , D ( k )) ≃ D ( k [ t ])and by i ne the left adjoint of D ( G T n ) ≃ H (Γ T n , F T n ) res e −−→ Fun( S , D ( k )) ≃ D ( k [ t ]) . The induction assumption implies that i e and i ne are each equivalent to either i + e or i − e (by abuseof notation). If both superscripts, each + or − , in this description are identical, we choose T e = T , where T is as in the base case of the induction, otherwise we choose T e = id Fun( S , D ( k )) .We denote the (Γ T n , V T n ) parametrized perverse schober obtained from the modification of F T n by F ∗ T n . We now define F T n +1 as the gluing of F T and F ∗ T n . The parametrized perverse schober F B T n +1 is then given by the gluing of F B T and ( F ∗ T n ) B n .We denote by i B , e and i B n ,ne the left adjoints of D ( G B T ) ≃ H (Γ B T , F B T ) res e −−→ Fun( S , D ( k )) ≃ D ( k [ t ]) , respectively, D ( G B n T n ) ≃ H (Γ B n T n , F B n T n ) res e −−→ Fun( S , D ( k )) ≃ D ( k [ t ]) . The induction assumption again implies that i B , e and i B n ,ne are equivalent to either i + e or i − e (again by abuse of notation). We define ( F B n T n ) ∗ as the (Γ B n T n , V T n )-parametrized perverse schoberobtained from composing F B n T n ( v → e ), for each edge v → e in Exit(Γ B n T n ) with e an admissibleedge in S , by T if the superscripts of i B , e and i B n ,ne match and by id Fun( S , D ( k )) otherwise.The parametrized perverse schober ( F B n T n ) ∗ is obtained by first removing non-admissible externaledges from F T n and then fixing the ’correct’ signs, whereas the parametrized perverse schober( F ∗ T n ) B n is obtained by first fixing the ’correct’ signs and then removing non-admissible externaledges. We show below that there exists an equivalence of (Γ B n T n , V T n )-parametrized perverseschobers ( F ∗ T n ) B n ≃ ( F B n T n ) ∗ . e now proceed with the computation of H (Γ B T n +1 , F B T n +1 ). Note that H (cid:0) Γ B n T n , ( F ∗ T n ) B n (cid:1) ≃ H (cid:0) Γ B n T n , F B n T n (cid:1) . Using the induction assumption we thus find equivalences of ∞ -categories H (cid:0) Γ B n T n , (cid:0) F B T n +1 | Exit(Γ Bn T n ) (cid:1)(cid:1) = H (cid:0) Γ B n T n , ( F ∗ T n ) B n (cid:1) ≃ ψ ( C B n T n ) , H (cid:0) Γ B T , (cid:0) F B T n +1 | Exit(Γ B T ) (cid:1)(cid:1) = H (cid:0) Γ B T , F B T (cid:1) ≃ ψ ( C B T ) . A standard result on the decomposition results of colimits, see [Lur09, 4.2.3.10], shows thatglobal sections commute with gluing in the sense that there is a pullback diagram in P r L asfollows. H (cid:0) Γ B T n +1 , F B T n +1 (cid:1) H (cid:0) Γ B n T n , ( F ∗ T n ) B n (cid:1) H (cid:0) Γ B T , F B T (cid:1) Fun( S , D ( k )) × s y The left adjoint diagram is equivalent to the image under ψ of the following homotopy pushoutdiagram in dgCat k . k [ t ] ∐ s C B n T n C B T C α n α p i C (57)The dg-functors α and α n are on the component of k [ t ] ∐ s indexed by a given edge e of eachof the form i + e or i − e (again by abuse of notation); we denote this component by i e and i ne ,respectively. We show below that α and α n are cofibrations. The diagram (57) is thus alsopushout. We now describe C . The number of objects of C is | ( Q B n T n ) | + | ( Q B T ) | − s . Themorphism are generated by the edges of Q B n T n and the edges of Q B T and for each edge e in S twoendomorphisms ( l ′ e, ) , ( l ′ e, ) n : e → e in degree 2 and one endomorphism l ′ e, : e → e in degree1. We find d (( l ′ e, ) ) = i e ( t ) − X ( f,j ) ,f in T p e [ a f,j , a ∗ f,j ] p e and d (( l ′ e, ) n ) = i ne ( t ) − X ( f,j ) ,f in T n p e [ a f,j , a ∗ f,j ] p e . Using the equivalence ( F ∗ T n ) B n ≃ ( F B n T n ) ∗ , we find i e ( t ) = − i ne ( t ) = ± l ′ e, . It follows that d ( − ( l ′ e, ) ) − ( l ′ e, ) n ) = X ( f,j ) ,f in T n +1 p e [ a f,j , a ∗ f,j ] p e is a boundary. We deduce the existence of a quasi-equivalence C B T n +1 → C . This shows the firstpart of the induction step.Let e be an admissible boundary edge of T n +1 . Assume that e lies in T n . The left adjoint ofthe restriction functor to e factors as D ( k [ t ]) ψ ( i ± e ) −−−−→ ψ ( C B n T n ) ψ ( i C ) −−−−→ ψ ( C ) ≃ ψ ( C B T n +1 )and thus equivalent to ψ ( i ± e ). If e lies in T , we can argue analogously. This shows the secondpart of the induction step, completing the induction. F ∗ T n ) B n and ( F B n T n ) ∗ are equivalent To describe an equivalence between ( F ∗ T n ) B n and ( F B n T n ) ∗ we consider the following localpicture of F ∗ T n at at an edge in S . Fun( S , D ( k ))Fun( S , D ( k )) V f ∗ Fun( S , D ( k )) T e ◦ ρ i ρ i +1 ρ i +2 (58)Here i, j are considered modulo 3. We assume that the target of the edge in Exit(Γ T n )labelled by T e ◦ ρ i lies in S , i.e. is being glued and that atleast one of the other two edges is notadmissible and external. We begin with the case where there is exactly one not admissible edge.Assume that it is the upper edge, i.e. the target of the edge in Exit(Γ T n ) labelled by ρ i +2 . UsingProposition 3.11 we can assume that ρ i +2 = ρ . Using Lemma 4.24, we obtain the followingdiagram of ∞ -categories. V f ∗ S , D ( k )) V f ∗ Fun( S , D ( k ))Fun( S , D ( k )) y ρ T e ◦ ρ T e ◦ ρ ρ ρ (59)The part of the above diagram containing V f ∗ , ̺ and T e ◦ ̺ is a local depiction of ( F ∗ T n ) B n .Using the twist functor T − V f ∗ , we find a natural equivalence between diagram (59) and thefollowing diagram. V f ∗ S , D ( k )) V f ∗ Fun( S , D ( k ))Fun( S , D ( k )) y ρ T ◦ T e ◦ ρ [1] T e ◦ ρ ρ ρ The left adjoints of the restriction maps res e : ψ ( C ∅ T n ) → D ( k [ t ]) and res B n e : ψ ( C B n T n ) → D ( k [ t ]) factor as D ( k [ t ]) ≃ Fun( S , D ( k )) ladj( T e ◦ ̺ ) −−−−−−−→ V f ∗ → ψ ( C ∅ T n ) , respectively D ( k [ t ]) ≃ Fun( S , D ( k )) ladj( T ◦ T e ◦ ̺ [1]) −−−−−−−−−−→ V f ∗ → ψ ( C B n T n ) . ote that ladj( T e ◦ T ◦ ̺ [1]) = ς ◦ T e and ladj( T e ◦ ̺ ) = ς ◦ T e and are thus both modelledby i + e if T e = id Fun( S , D ( k )) and i − e if T e = T . This shows that locally at e the parametrizedperverse schobers ( F ∗ T n ) B n and ( F B n T n ) ∗ are naturally equivalent (with a natural equivalencewhich restricts to the identity on both Fun( S , D ( k )). If the non-admissible edge is the loweredge in (58), i.e. the target of the edge in Exit(Γ T n ) labelled by ̺ i +1 , we argue analogously andobtain the following diagram. V f ∗ S , D ( k )) V f ∗ Fun( S , D ( k ))Fun( S , D ( k )) y ρ T e ◦ ρ T e ◦ ρ ρ ρ Using the twist functor T V f ∗ , we find a natural equivalence between diagram (59) and thefollowing diagram. V f ∗ S , D ( k )) V f ∗ Fun( S , D ( k ))Fun( S , D ( k )) y T ◦ ρ [1] T e ◦ ρ T e ◦ ρ ̺ ρ Arguing then as above and using that ladj( ̺ ) = ς is modelled by i − and ladj( ̺ ) = T ◦ ̺ is modelled by i − , we again find the parametrized perverse schobers ( F ∗ T n ) B n and ( F B n T n ) ∗ to benaturally equivalent.If two edges are not admissible, then locally the perverse schobers ( F ∗ T n ) B n and ( F B n T n ) ∗ areof the form Fun( S , D ( k )) T ′ e ◦ T e ◦ ̺ ←−−−−−− V f ∗ where in each case either T ′ e = T or T ′ e = id Fun( S , D ( k )) . The resulting perverse schobers arelocally equivalent, as follows from the fact the left duals are locally modelled by i +1 = i − . Theconstructed local equivalences between ( F ∗ T n ) B n and ( F B n T n ) ∗ glue together to an equivalence ofparametrized perverse schobers. α n and α are cofibrations We now show that α n is a cofibration, it follows by the same argument that α is alsoa cofibration. We first note that if T n consists of a single ideal triangle and 1 ≤ s ≤ T n , the dg-functor α n is a cofibration. This can be directly checkedby verifying the lifting property.We now continue with an induction on the number of ideal triangles of the triangulation.For the induction step, consider the case that the boundary edges in S are incident to at mosttwo ideal triangles T ′ and T ′′ of T n with 1 ≤ s ′ ≤ T ′ and 1 ≤ s ′′ ≤ T ′′ . We can decompose T n into two subtriangulations T and T from which T n is obtained ia the gluing along t many edge, such that T ′ ⊂ T and T ′′ ⊂ T . Denote B = B n ∩ (Γ T ) and B = B n ∩ (Γ T ) . We find the following commutative diagram in dgCat k . ∅ k [ t ] ∐ s ′ k [ t ] ∐ t C B T k [ t ] ∐ s ′′ k [ t ] ∐ s C B T C B n T n α n It is apparent that the front and back face of the above cube are pushout squares and that alledges except α n are cofibrations. It follows that α n is a cofibration.To complete the induction step, we now consider the case where the boundary edges in S are incident to three ideal triangles T ′ , T ′′ and T ′′′ . We again decompose T n into threesubtriangulations T , T and T such that T ′ ⊂ T , T ⊂ T and T ′′′ ⊂ T ′′ . We can assumewithout loss of generality that T and T ′ are connected in T n . Denote the subtriangulation of T n consisting of T and T by T . Denote B = B n ∩ (Γ T ) . As a special case of the aboveargument in the case of two subtriangulations of T n , where T n = T and s ′ = s ′′ = 1, we findthat the the restriction of α n to k [ t ] ∐ → C B T to be a cofibration. Applying the same argumentagain, we obtain that α n is a cofibration, completing the induction.We now modify the perverse schober F T to also describe the perfect and finite modules overthe relative Ginzburg algebra G T . Notation 6.4.
We denote• by D ( k ) perf ⊂ D ( k ) the full subcategory spanned by perfect complexes of k -vector spaces.• by Fun( S , D ( k )) perf ⊂ Fun( S , D ( k )) the full subcategory spanned by compact objects.Note that Fun( S , D ( k ) perf ) is contained in the full subcategory of Fun( S , D ( k )) spannedby compact objects, denoted Fun( S , D ( k )) perf .• by ( f ∗ ) perf : D ( k ) perf ←→ Fun( S , D ( k )) perf : f perf ∗ and ( f ∗ ) fin : D ( k ) perf ←→ Fun( S , D ( k ) perf ) : f fin ∗ the restrictions of the adjunction f ∗ : D ( k ) ↔ Fun( S , D ( k )). For the well-definednessof f perf ∗ , note that f ∗ preserves filtered colimits, so that f ∗ carries compact objects tocompact objects.Adapting the construction in the proof of Theorem 6.2, we define the (Γ T , V T )-parametrizedperverse schober ( F T ) fin : Exit(Γ T ) → Stby replacing in F T at each singular vertex the spherical functor f ∗ by ( f ∗ ) fin and the (Γ B T , V T )-parametrized perverse schober ( F B T ) perf : Exit(Γ B T ) → Stby replacing in F B T at each singular vertex the spherical functor f ∗ by ( f ∗ ) perf . Note that( F B T ) perf , ( F T ) fin take values in the ∞ -category St idem ⊂ St of idempotent complete, stable ∞ -categories. roposition 6.5. Consider the morphism F fin T → F T in P (Γ T , V T ) given by the pointwise in-clusion and the morphism ( F B T ) perf → F B T in P (Γ B T , V T ) given pointwise by Ind -completion.Passing to global sections with support on B c , respectively colimits, yields the following commu-tative diagram of ∞ -categories colim St idem D R ( F B T ) perf colim P r L D R F B T H B c (Γ T , F T ) H B c (Γ T , F fin T ) D ( G B T ) perf D ( G B T ) D ( G B T ) D ( G B T ) fin ≃ ≃ ≃ ≃ ≃ id where • D ( G B T ) perf ⊂ D ( G B T ) denotes the full subcategory spanned by the compact objects and • D ( G B T ) fin ⊂ D ( G B T ) denotes the full subcategory spanned by the modules with finite totalhomological dimension.In particular, we obtain that the finite G B T modules can be characterized as the coCartesian sec-tions of the Grothendieck construction Γ( F T ) → Exit(Γ T ) whose pointwise values lie in N ( f ∗ ) fin and V i ( f ∗ ) fin and which vanish on all edges in B .Proof. It follows from discussion on the computation of colimits in St idem in Section 2.1 thatthe colimit of D R ( F B T ) perf describes the ∞ -category of compact G B T -modules.A G B T -module M is by definition finite if and only if RHom G B T ( G B T , M ) ∈ D ( k ) is a finite k -module. Note that G B T = L e p e G B T , where the sum runs over all edges of T not lying in B .Using Proposition 6.8, we thus obtain that M is finite if and only if RHom G B T ( p e G B T , M ) ≃ i ∗ ev e ( M ) ∈ D ( k ) is a finite k -module for all e . The latter is fulfilled if and only if the pointwisevalues of the coCartesian section corresponding to M lies in N ( f ∗ ) fin and V i ( f ∗ ) fin .Proposition 6.5 shows that we can also describe the perfect derived category of G B T via asuitable colimit. Our reason for the formulation of Proposition 6.5 in terms of the parametrizedperverse schober F B T in Proposition 6.5 is purely technical: we are not aware of a good descriptionof the analogue of the global sections with support in B c in the dual perspective on perverseschobers; we thus need to consider to the auxiliary object F B T . The dual perspective on perverseschobers and the description of their global sections as colimits can be applied to algebraicdescriptions of these. Proposition 6.5 highlights a feature of the limit description of globalsections of perverse schobers: we can very explicitly describe the (finite) objects therein. Wefurther illustrate this in examples in Section 6.3.The ∞ -category D ( G B T ) fin cannot be described via the global sections of the dual of a (sen-sible) parametrized perverse schober, meaning it does not arise via the gluing constructionas a colimit. It thus appears that the gluing construction of D ( G ( Q ◦ T , W ′ T )) of Corollary 6.3does not descent to the 2-CY cluster category, as defined by Amiot as the Verdier quotient C ( Q ◦ T ,W ′ T ) := D ( G ( Q ◦ T , W ′ T )) perf / D ( G ( Q ◦ T , W ′ T )) fin . In this section we illustrate the computation of the proof of Theorem 6.2 in the case of theonce-punctured torus and the unpunctured square. The Verdier quotient is given in the context of ∞ -categories by the cofiber in St idem of the inclusion functor,see [BGT13, Section 5.1]. xample 6.6. Let S be the once punctured torus and consider the ideal triangulation T of S consisting of two triangles glued together at all three edges. Using the graphical Notation 4.6,the dual ribbon graphs with singularities (Γ T , V T ) can be depicted as follows (the crossing hasto do with the cyclic orderings of the halfedges induced by the orientation of S ). × × Consider the dg-category D with three objects x, y, z , freely generated by the followingmorphisms. yx z a ∗ ba c ∗ b ∗ c The morphisms are in degrees | a | = | b | = | c | = 0 and | a ∗ | = | b ∗ | = | c ∗ | = 1. The differentialis determined by d ( a ∗ ) = cb, d ( b ∗ ) = ac, d ( c ∗ ) = ba . Note that the dg-category D is Moritaequivalent to the deformed graded path algebra of the double of the quiver consisting of a single3-cycle. To describe the associated Ginzburg algebra consider the following span of dg-categories,see Notation 5.11 and Propositions 5.9 and 5.12 for the notation. k [ t ] ∐ D D i − ,i +2 ,i − )( i +1 ,i − ,i +3 ) (60)Informally, the above span describes a gluing of two copies of the dg-category D at all verticeswith matching orientations. To compute the homotopy colimit of (60), we consider the cofibrantreplacement of the diagram (60), which consists in replacing D with the following dg-category. yx z a ∗ bl y l y a c ∗ l x l x b ∗ c l z l z (61)Here | l x | = | l y | = | l z | = 1 and | l x | = | l y | = | l z | = 2 and d ( l x ) = l x − ( cc ∗ − a ∗ a ) ,d ( l y ) = l y − ( aa ∗ − b ∗ b ) ,d ( l z ) = l z − ( bb ∗ − c ∗ c ) . The colimit of the cofibrant replacement of (60) is up to quasi-isomorphism the freely generated g-category of the following form. yx z a ∗ b a ∗ b l y a c ∗ l x a c ∗ b ∗ c l z b ∗ c A direct computation shows that the differentials match the differentials of the Ginzburg algebra G T ≃ G ( Q T , W ′ T ) of the quiver ·· · b b a a c c (62)with potential W ′ T = c b a + c b a .The above computation relates to the perverse schober F T in the following way. Denote by T the functor Fun( S , D ( k )) (35) ≃ D ( k [ t ]) ϕ ∗ −−→ D ( k [ t ]) (35) ≃ Fun( S , D ( k )), where ϕ ∗ is the pullbackfunctor along the morphism of dg-algebras ϕ : k [ t ] t t −−−−−→ k [ t ]. The (Γ T , V T )-parametrizedperverse schober F T : Exit(Γ T ) → St is given by the following diagram. V f ∗ N f ∗ N f ∗ N f ∗ V f ∗ T ◦ ̺ T ◦ ̺ T ◦ ̺ ̺ ̺ ̺ (63)The limit of this diagram is equivalent to the colimit in P r L of the right adjoint diagram. Astandard argument, for example using the decomposition of colimits [Lur09, 4.2.3.10], showsthat the colimit of the right adjoint diagram is equivalent to the colimit of the following spanin P r L . N × f ∗ V f ∗ V f ∗ ( ς ,ς ,ς )( ς ,ς ,ς ) (64)The above span is modelled in terms of dg-categories, i.e. equivalent to the image under ψ from(11), by the span (60), so that the colimit of (64) in P r L is equivalent to the image under ψ of the homotopy colimit of (60) and thus equivalent to the derived category of the Ginzburgalgebra G T . Finally, we wish to emphasize that to get the correct signs in the differential of theGinzburg algebra, the appearance of the autoequivalence T in diagram (63) is crucial.In next example of the unpunctured square the surface has a non-empty boundary, so thatassociated the relative Ginzburg algebra G T contains more information than the non-relativeGinzburg algebra G ( Q ◦ T , W ′ T ). xample 6.7. Let S be the unpunctured square and T the ideal triangulationconsisting of two not self-folded ideal triangles. The dual ribbon graph with singularities (Γ T , V T )can be depicted as follows. × × The right dual of the corresponding parametrized perverse schober F T : Exit(Γ T ) → St ismodelled by the following diagram in dgCat k . k [ t ] k [ t ] D k [ t ] D k [ t ] k [ t ] i + x i − z i + y i − y i − z i + x (65)To compute the homotopy colimit of the above diagram, we again consider the cofibrant re-placement, which consists of replacing D with the dg-category depicted in (61). The colimitof the cofibrant replacement of (65) is quasi-equivalent to the freely generated dg-category withfive objects, which can be depicted as follows. z x y y z c b ∗ l y c ∗ a c ∗ a b a ∗ a ∗ b c b ∗ (66)The nonzero differentials are given by d ( l y ) = c c ∗ + c c ∗ − a ∗ a − a ∗ a and for i = 1 , d ( a ∗ i ) = c i b i , d ( b ∗ i ) = a i c i , d ( c ∗ i ) = b i a i . It follows that the dg-category (66) is Morita equivalent to the relative Ginzburg algebra G T associated to the ideal triangulation. he ∞ -category of global sections of F T with support on the sub-ribbon graph Γ ◦ T of internaledges can be described as the colimit of the diagram in P r L , D ( k [ t ]) V f ∗ V f ∗ f ! f ! (67)where f ! : D ( k ) → Fun( S , D ( k )) = V f ∗ is the left adjoint of the pullback functor f ∗ along f : S → ∗ . The diagram (67) is modelled by the following diagram in dgCat k k [ t ] k k φφ (68)where φ : k [ t ] t −−−→ k . The homotopy colimit of (68) is given by the polynomial algebra k [ t ]with | t | = 2, which is the Ginzburg algebra of the A -quiver.Finally, we wish to illustrate the gluing construction in the case that there are vertices ofvalency 2 in the ribbon graph. For that consider the ∞ -category of sections of F T with supporton the sub-ribbon graph Γ ′ of Γ, which we depict as follows. × × We can describe H Γ ′ (Γ , F T ) as the colimit of the following diagram in P r L . D ( k [ t ]) V f ∗ V f ∗ f ! ς (69)The ∞ -category V f ∗ is modelled by the dg-category D , which is the freely generated dg-categorywith two objects y, z and morphisms of the following form, y z bb ∗ with | b | = 0 , | b ∗ | = 1, see Proposition 5.9. The colimit of (69) is thus equivalent to the ∞ -category of modules over the path algebra of the graded quiver y z bb ∗ l z with differential d ( l z ) = bb ∗ . .3 Spherical indecomposable and projective, indecomposable mod-ules Let T be an ideal triangulation of a surface and B ⊂ (Γ T ) ∂ a subset of the boundary edges of T and B c = (Γ T ) \ B . Assume that the commutative ring k is a field. The ∞ -category D ( G T )admits• for each edge e of T not lying in B an indecomposable object S e which is 3-spherical if e is an interior edge and exceptional if e is a boundary edge.• for each interior vertex v of T an indecomposable object C v with the chain complex un-derlying the k -linear endomorphism algebra equivalent to H ∗ ( S × S , k ).• for each edge e of T not lying in B an indecomposable object P Be = p e G B T , the correspond-ing G B T -module of which is projective.The goal of this section is to identify the above listed modules in H B c (Γ T , F T ) ≃ D ( G B T ). If B = Γ ∂ , the finite derived 1-category D ( G B T ) fin = Ho D ( G B T ) fin is a full subcategory of an(untwisted) Fukaya-category, see [Smi15, Section 5.4]. In this description the S e correspond toLagrangian matching spheres and the C v correspond to Lagrangian embeddings of S × S .We can use the limit descriptions of the ∞ -categories of global sections to geometricallydescribe the modules objects S e and C v , as coCartesian sections of G ( F B T ). We begin withthe case where e is an interior edge. This description can be seen as the categorical analogueof the description as Lagrangian matching spheres of the corresponding objects in the Fukayacategories appearing in [Smi15]. Consider an edge e of the ideal triangulation T , connecting twoadjacent ideal triangles labelled a, b . Locally at e , we find F T to be up to natural equivalenceof the following form: V f ∗ ̺ −−→ Fun( S , D ( k )) T ◦ ̺ ←−−− V f ∗ We observe that there is a pullback diagram in Cat ∞ . D ( k ) V f ∗ S n − , D ( k )) × ι y ( ̺ ,̺ ) (70)We thus find a coCartesian section of the Grothendieck construction Γ( F T ), see Section 2.1,lying in G B T ≃ H B c (Γ T , F T ) ⊂ H (Γ T , F T ), which is locally at e of the form ι ( k ) ! −−→ f ∗ ( k ) ! ←−− ι ( k ) , where ι : D ( k ) → V f ∗ denotes the inclusion of the first component of the semiorthogonaldecomposition. and vanishes otherwise. This coCartesian section describes the objects S e .Suppose now that e is an edge lying on the boundary and not in B , incident to an ideal triangle a . Locally at e , F T is given up to natural equivalence as follows:Fun( S , D ( k )) ̺ ←−− V f ∗ Using again the pullback diagram (70), we find that there exists a coCartesian section of Γ( F T )of the form f ∗ ( k ) ! ←−− ι ( k ) , describing the desired object S e in G B T ⊂ H (Γ T , F T ). Note that S e is an exceptional object,i.e. its k -linear endomorphism algebra is given by k . Let v be an interior vertex of T . Locally t the circle around v , F T is up to natural equivalence of the following from.Fun( S , D ( k ))Fun( S , D ( k )) V f ∗ Fun( S , D ( k ))Fun( S , D ( k )) V f ∗ V f ∗ Fun( S , D ( k ))Fun( S , D ( k )) . . . . . . ̺ ̺ ̺ ̺ ̺ ̺ ̺ ̺ ̺ The corresponding indecomposable object C v is given by the coCartesian section which canbe depicted as follows, where ι is the inclusion of the third component of the semiorthogonaldecomposition of V f ∗ . 0 f ∗ ( k ) ι ( f ∗ ( k )) f ∗ ( k )0 ι ( f ∗ ( k )) ι ( f ∗ ( k )) 0 f ∗ ( k ) . . . . . . ̺ ̺ ̺ ̺ ̺ ̺ ̺ ̺ ̺ We now describe the projective, indecomposable objects P Be . Let e be an edge of T notlying in B . Denote by ev Be : H B c (Γ T , F T ) → Fun( S , D ( k )) the evaluation functor at thevertex e ∈ Exit(Γ T ). Using that limits and colimits in H (Γ T , F T ) are computed pointwise, itfollows that limits and colimits in H B c (Γ T , F T ) are also computed pointwise. We obtain thatthe functor ev Be preserves all limits and colimits and thus admits a left adjoint, denoted (ev Be ) ∗ .Consider further the left functor i ! : D ( k ) → Fun( S , D ( k )) appearing in Lemma 5.4. We showin Proposition 6.8 that the projective, indecomposable object P Be ∈ H B c (Γ T , F T ) is equivalentto (ev Be ) ∗ i ! ( k ). Proposition 6.8.
Let T be an ideal triangulation of a surface, B a subset of the set of boundaryedges and B c = (Γ T ) \ B and let e ∈ B c . The left adjoint (ev Be ) ∗ i ! of the functor i ∗ ev Be : H B c (Γ T , F T ) → D ( k ) maps k ∈ D ( k ) to p e G B T ∈ D ( G B T ) ≃ H B c (Γ T , F T ) .Proof. We prove the statement via induction on the number of ideal triangles of T . Let T bethe ideal triangulation consisting of a single not self-folded triangle. Proposition 5.12 showsthat (ev Be ) ∗ i ! is equivalent to the image under ψ of the functor of dg-categories α : k → D −| B | determined by mapping the object of k to one of the objects of D −| B | . It follows that p e G B T ≃ ψ ( α )( k ) ≃ (ev Be ) ∗ i ! ( k ), as desired. We leave the case of the triangulation consisting of a singleself-folded triangle to the reader.Suppose the statement has been shown for all ideal triangulations T with at most n idealtriangles. The setup is as in the induction step in the proof of Theorem 6.2: we consider anideal triangulation T n +1 with n + 1 ideal triangles obtained via the gluing of ideal triangulations n and T with n , respectively, 1 ideal triangles along s boundary edge. Let B n = (Γ T n ) ∩ B and B cn = (Γ T n ) \ B n . If e is an edge in T n not lying in B n , the functor (ev Be ) ∗ i ! : D ( k ) → H B c (Γ T n +1 , F T n +1 ) ≃ H (Γ B T n +1 , F B T n +1 ) ≃ D ( G B T n +1 ) factors as D ( k ) (ev Bne ) ∗ i ! −−−−−−→ H B cn (Γ T n , F T n ) ≃ H (Γ B n T n , F B n T n ) α −−→ H (Γ B T n +1 , F B T n +1 ) ≃ D ( G B T n +1 ) , where α is modelled by an inclusion C B n T n → C B T n +1 . Using the induction assumption andLemma 2.23, it follows that (ev Be ) ∗ i ! ( k ) is given by p e G B T n +1 ∈ D ( G B T n +1 ) ≃ H B c (Γ T n +1 , F T n +1 ).If e is an edge of T , we can argue analogously. The statement thus follows. Given an ideal triangulation T of a surface we can produce a new triangulation T ′ by a flip ofany not self-folded edge e , locally by applying the following operation on T :flipAs shown in [LF09] this corresponds on the associated quivers with potential to the quivermutation at the vertex of the quiver corresponding to the edge e . It is shown in [Kel11, Section7.6] that there exists an associated equivalence between the derived categories of the 3-CYGinzburg algebras G ( Q ◦ T , W ′ T ) and G ( Q ◦ T ′ , W ′ T ′ ). The goal of this section is to associate toeach flip an equivalence between the derived ∞ -categories of the relative Ginzburg algebras G B T and G B T ′ for B ⊂ (Γ T ) ∂ , thus partially extending and partially recovering the combined resultof [LF09], [Kel11].From the perspective of dual ribbon graphs with singularities, a flip of a triangulation relatesthe two ribbon graphs Γ T and Γ T ′ , which locally differ as follows: × × flip ! ×× (71)The flip (71) can be described by the pair of spans of contractions of ribbon graphs withsingularities (72) and (73) below. × × c ←−− × · · × c −−→ × · × (72) × · × c ←−− · ×× · c −−→ ×× (73) enote by T the autoequivalence where ϕ : k [ t ] t t −−−−−→ k [ t ]. Using the pushforward functorsof Proposition 4.26 along the contractions of ribbon graphs in (72) and (73), we find the followingequivalences of parametrized perverse schobers. f ∗ f ∗ ̺ ̺ ( ̺ ,T ◦ ̺ ) ̺ ̺ ≃ (74) ≃ f ∗ f ∗ ̺ ̺ ( ̺ ,̺ ) T ◦ ̺ T ◦ ̺ ( c ) ∗ ←−−−− f ∗ O N f ∗ O N f ∗ f ∗ ̺ ̺ ( ̺ ,̺ ) ( ̺ ,̺ ) ( ̺ ,̺ ) T ◦ ̺ T ◦ ̺ ≃ (75) ≃ f ∗ O N f ∗ O N f ∗ f ∗ ̺ [1] ̺ ( ̺ ,̺ [1]) ( ̺ ,̺ ) ( ̺ ,̺ ) T ◦ ̺ T ◦ ̺ ( c ) ∗ −−−−→ f ∗ O N f ∗ f ∗ ̺ [1] ̺ ( ̺ ,̺ [1]) ( ̺ ,̺ ) T ◦ ̺ T ◦ ̺ ≃ (76) ≃ f ∗ O N f ∗ f ∗ ̺ [1] ̺ ( ̺ ,̺ [3]) ( ̺ [2] ,̺ ) T ◦ ̺ T ◦ ̺ [2] ( c ) ∗ ←−−−− O N f ∗ f ∗ f ∗ O N f ∗ ̺ [1]( ̺ [2] ,̺ ) T ◦ ̺ ̺ ( ̺ ,̺ [3]) ( ̺ ,̺ ) T ◦ ̺ [2] ≃ (77) ≃ O N f ∗ f ∗ f ∗ O N f ∗ ̺ [2]( ̺ [2] ,̺ ) T ◦ ̺ ̺ ( ̺ ,̺ [3]) ( ̺ [1] ,̺ ) T ◦ ̺ [3] ( c ) ∗ −−−−→ f ∗ f ∗ ̺ [2] T ◦ ̺ [2] ̺ [3]( ̺ [1] ,̺ ) T ◦ ̺ [3] ≃ (78) ≃ f ∗ f ∗ T ◦ ̺ ̺ ̺ ( ̺ ,T ◦ ̺ ) T ◦ ̺ (79) ach of the above equivalences of parametrized perverse schober is nontrivial only at one ortwo singular vertices with label 0 N f ∗ , where it is given by the twist functor T V n N f ∗ of Lemma 3.8,see also Proposition 3.11, except for the equivalence between the parametrized perverse schoberin (74) and the left parametrized perverse schober in (75) and the equivalence between the rightparametrized perverse schober of (78) and the parametrized perverse schober of (79). The formeris nontrivial only at the lower vertex labelled f ∗ , where it is given by the autoequivalence ǫ of V f ∗ which restricts on both the components Fun( S , D ( k )) of the semiorthogonal decompositionto T and on the component D ( k ) of the semiorthogonal decomposition to the identity functor.The latter equivalence of parametrized perverse schobers is nontrivial at three vertices of theexit path category corresponding to the two singular vertices and the edge connecting them. Atthe lower singular vertex, the equivalence is given by [3], at the upper singular vertex by ǫ ◦ [2]and at the vertex corresponding to the edge by [ − T , T of a surface, related by a flip of an interior edge e ∈ T which is not the self-folded edge of a self-folded triangle. We also assume for now thatneither e nor its flip e ′ are the outer edge of a self-folded triangle. Consider further the (Γ T i , V T i )-parametrized perverse schobers F T i , for i = 1 ,
2. By replacing F T with an equivalent perverseschober, we can assume that the diagram underlying F T restricts to the diagram underlyingthe parametrized perverse schober of (74). We can thus describe F T as the gluing of theparametrized perverse schober of (74) and its complement in F T . We now observe that F T can be chosen so that it is the gluing of the complement of the parametrized perverse schober(74) in F T and the parametrized perverse schober (79). This rests on the observation, that theleft adjoints of ̺ and T ◦ ̺ are modelled by i +3 and i +2 , respectively, the signs of which agree.We can thus glue the complement of the parametrized perverse schober (74) in F T with theparametrized perverse schobers (74)-(79) along their stops and use Proposition 4.26 to obtainan equivalence of ∞ -categories µ e : H (Γ T , F T ) → H (Γ T , F T )which we call the mutation equivalence at e . If either e or e ′ is the outer edge of a self-foldedtriangle, the construction of µ e can be slightly modified by additionally gluing in a parametrizedperverse schober of the form 0 N f ∗ at the two of the stops corresponding to the self-folded edge. We thus obtain a mutationequivalence µ e for each edge e which is not the self-folded edge of a self-folded triangle.The mutation equivalence µ e clearly maps coCartesian section of the Grothendieck construc-tion of F T vanishing at a given subset B of the set of boundary edges to coCartesian sectionswith the same vanishing property and thus restricts to an equivalence of ∞ -categories µ e : H B c (Γ T , F T ) → H B c (Γ T , F T ) . Proposition 6.9.
Let T , T be two ideal triangulations of a surface, differing by a flip of aninterior edge e ∈ T to e ′ ∈ T . Let B ⊂ (Γ T ) ∂ = (Γ T ) ∂ be a subset of the sets of externaledges.1. Let I B be the set of edges of T not lying in B and incident to e ′ which follow the edge e ′ at one of the two singular vertices in the paracyclic order. There exist an equivalence in H (Γ T , F T ) µ e ( P Be ) ≃ cof (cid:0) P Be ′ β Be −−→ M f ′ ∈ I B P Bf ′ (cid:1) , (80) to the cofiber of a canonical edge β Be . . Let f = e be an edge of T not lying in B and f ′ the corresponding edge of T . Thereexists an equivalence in H (Γ T , F T ) µ e ( P Bf ) ≃ P Bf ′ . Proof.
We begin by showing statement 1. The idea of the proof is to trace through the equiv-alences on global sections induced by (74)-(79) and describe the composition of the inverses ofthese equivalences with the evaluation at a point functor H B (Γ T , F T ) ev Be −−→ Fun( S , D ( k )) i ∗ −→ D ( k ). Passing to left adjoints and evaluating at k ∈ D ( k ) yields the image of P Be under µ Be .We begin with the cases where B is such that I B = I ∅ and describe afterwards how the argu-ment can be adapted to treat arbitrary B . We denote the complement of the left parametrizedperverse schober of (74) in F B T by F c .We denote the gluing of F c with the parametrized perverse schober• on the right of (76) by G .• on the left of (77) by G .• on the right of (77) by G .• on the left of (78) by G .The evaluation of G at the central nonsingular vertex yields the ∞ -category V N f ∗ ≃ { Fun( S , D ( k )) , Fun( S , D ( k )) , Fun( S , D ( k )) } . A direct computation shows that the composite of the equivalence lim G ≃ H B (Γ T , F T ) withev Be is given by the composite functor R of the evaluation at the central nonsingular vertex(labelled V N f ∗ ) with the restriction functor to the second component of the semiorthogonaldecomposition of V N f ∗ . Precomposing the functor R : lim G → Fun( S , D ( k )) with theequivalence lim G ≃ lim G yields the functor R given by the composite of the evaluation func-tor to the central nonsingular vertex (labelled V N f ∗ ) with the functor cof , [1] which denotesthe composite of the restriction functor to the first and third component of the semiorthogonaldecomposition with the suspension of the cofiber functor. Denote the diagonal edge of the mid-dle ribbon graph of (73) by e ′′ , adjacent to the upper vertex denoted v and the lower vertexdenoted v . Note that ev Be ′′ ≃ ̺ ◦ ev Bv ≃ ̺ ◦ ev Bv . Further, there exist canonical naturaltransformations a : ̺ [ −
1] = π −→ π = ̺ , given at a vertex x α −→ y ∈ V N f ∗ = { Fun( S , D ( k )) , Fun( S , D ( k )) } by the edge α and acanonical natural transformation b : ̺ = cof , −→ π [1] = ̺ , given by b = a ◦ T V N f ∗ [ − R withthe equivalence lim G ≃ lim G induced by the contraction c yields the functor R : lim G → Fun( S , D ( k )) given by the suspension of the cofibercof (cid:0) ̺ [ − ◦ ev Bv ⊕ ̺ ◦ ev Bv ( a ◦ ev Bv ,b ◦ ev Bv ) −−−−−−−−−−→ ev Be ′′ (cid:1) [1]in the stable ∞ -category Fun(lim G , Fun( S , D ( k )). The composition R of R with lim G ≃ lim G yields the functor given by the suspension of the cofibercof (cid:0) ̺ ◦ ev Bv ⊕ ̺ [1] ◦ ev Bv → ev Be ′′ (cid:1) [1] ≃ cof (cid:0) ev Bf [ − ⊕ ev Bf [ − → ev Be ′′ (cid:1) [1] , here f and f denote the edges corresponding to the stops adjacent to v , respectively, v .Continuing this way, we see that the composite of the equivalence H B (Γ T , F T ) ≃ H B (Γ T , F T )with ev Be yields the functor R = cof (cid:0) ev Bf [ − ⊕ ev Bf [ − → ev Be ′ [ − (cid:1) [1] ≃ fib (cid:0) ev Bf ⊕ ev Bf → ev Be ′ (cid:1) . Using that i ∗ ◦ ev Bg is modelled by Hom G B T ( P Bg , -) for g = e ′ , f , f , see Proposition 6.8, weobtain that the composite i ∗ ◦ R is modelled byHom G B T (cid:0) cof (cid:0) P Be ′ β B −−→ P Bf ⊕ P Bf (cid:1) , - (cid:1) for some edge β B and the left adjoint thus maps k ∈ D ( k ) to fib (cid:0) L f ′ ∈ I P Bf ′ β B −−→ P Be ′ (cid:1) , showingstatement 3 in the case I B = I ∅ . If I B = I ∅ , we can argue analogously as follows. The vertex v , the vertex v or both v and v are adjacent to external edges in B and one removes ev v ,ev v or ev v and ev v , respectively, from the above computation and replaces V N f ∗ by V N f ∗ , V N f ∗ or V N f ∗ , respectively.Statement 2 can be approached like statement 1 but is immediate, because the respectiveedges corresponding to the the projective objects are not affected by (74)-(79). Remark 6.10.
The formulas in Proposition 6.9 for the images of the the projective objectsunder µ e recover the formulas given in the context of completed (non-relative) Ginzburg algebrasin [KY11]. In the context of completed Ginzburg algebras, it is noted in Theorem 7.4 in [Kel12]that there exist two mutation equivalences for each vertex i of the quiver, which differ by thespherical twist of the spherical object associated to i . In terms of our construction, we can alsoproduce a second mutation equivalence µ ′ e which differs pointwise by the spherical twist around S e by replacing the spans of ribbons graphs (72) and (73) by the spans × × ←− ×· ·× −→ ×·××·× ←− ×··× −→ ×× and adapting the construction of the mutation equivalence µ e .We expect that there exists a natural equivalence µ e ≃ T S e ◦ µ ′ e , where T S e denotes the twistfunctor of the spherical adjunction induced by the spherical object S e . Further directions
In this section we show that the equivalence class of the parametrized perverse schober F T constructed in Theorem 6.2 not depend on any of the choices made in its construction. Beforewe can state a precise result, we briefly discuss combinatorial models for spin surfaces, following[DK15]. Definition 7.1.
Consider a pair ( S , V ) consisting of an oriented marked surface S , with the setof marked points denotes by M , and a finite set of singular points V ⊂ S ◦ \ M . Let (Γ , V ′ ) bea graph with singularities with an embedding f : | Exit(Γ) | → S \ M and consider the inducedribbon graph with singularities (Γ , V ′ ), see Remark 4.5. Let S = ∂ | Exit(Γ) | be the boundaryof | Exit(Γ) | in S \ M . We call f (or by abuse of notation Γ) a spanning graph for S if1. the embedding f is a homotopy equivalence,2. f induces a homotopy equivalence S → ∂ S \ M .If furthermore3. f restricts to a bijection between V ′ and V .then we call f (or by abuse of notation (Γ , V ′ )) a spanning graph with singularities for ( S , V ). Lemma 7.2.
Let S be a marked surface with an ideal triangulation T . Let V be a set containingan interior point of each ideal triangles of T . The dual ribbon graph with singularities (Γ T , V T ) is a spanning graph with singularities for ( S , V ) . Definition 7.3.
Let (Γ , V ) be a ribbon graph with singularities.1. We define the incidence diagram I : Exit(Γ) → Set to be the functor determined by• I ( v ) = H( v ) for v ∈ Γ ⊂ Exit(Γ) ,• I ( e ) = { e , e } for an edge e ∈ Γ consisting of halfedges e , e (counted twice forexternal edges),• and assigning to a morphism v → e from a vertex v ∈ Γ to an incident edge e ∈ Γ with σ ( e ) = v the morphism of sets H( v ) → { e , e } , mapping e to e andH( v ) \{ e } to e .2. We define the relative incidence diagram I ∗ : Exit(Γ) → ∆ × Set to be the functordetermined by• I ( v ) = { } × H( v ) for v ∈ V ⊂ Γ ⊂ Exit(Γ) ,• I ( v ) = { } × H( v ) for v ∈ (Γ \ V ) ⊂ Exit(Γ) ,• I ( e ) = { } × { e , e } for an edge e ∈ Γ consisting of halfedges e , e ,• and assigning to a morphism ( i, v ) → (1 , e ), with i = 0 ,
1, with v ∈ Γ and e ∈ Γ with σ ( e ) = v the morphism { i } × H( v ) → { } × { v , v } , mapping e to e andH( v ) \{ e } to e .We now define the 2-cyclic category Λ . The definition is similar to the definition of theparacyclic category Λ ∞ , see Definition 3.5, with the difference that the cyclic automorphisms τ n satisfy the additional relation ( τ n ) n +1) = id [ n ] . Definition 7.4.
For n ≥
0, let [ n ] denote the set { , . . . , n } . The 2-cyclic category Λ has asobjects the sets [ n ]. The morphism in Λ are generated by morphisms• δ , . . . , δ n : [ n − → [ n ],• σ , . . . , σ n − : [ n ] → [ n − τ n : [ n ] → [ n ], ubject to the simplicial relations and the further relations( τ n ) n +1) = id [ n ] ,τ n δ i = δ i − τ n − for i > , τ n δ = δ n ,τ n σ i = τ n +1 σ i − for i > , τ n σ = σ n . Definition 7.5.
Let ( S , V ) be an oriented marked surface with singularities and (Γ , V ) a span-ning ribbon graphs with singularities. Denote by N ⊂ Set the full subcategory spanned byobjects of the form [ n ] with n ≥
0. Choose a natural equivalence ν : I ≃ = ⇒ ˜ I , where˜ I : Exit(Γ) → N , which respects the cyclic orders on I ( x ) for x ∈ Exit(Γ) and on ˜ I ( x ) = [ | I ( x ) | − S \ ( M ∩ S ◦ ) (or on Γ) is a lift ˜ I Λ Exit(Γ) N ˜ I ˜ I of the diagram ˜ I . Remark 7.6.
In the definition of spin structure on a surface we deviate from [DK15] to keepthe exposition more direct and better applicable. While a spin structure in [DK15] consists ofsuitable Z -torsors at every trivalent vertex of the ribbon graph and Z -torsors at every edge ofthe ribbon graph, we include a choice of base point for each torsor, encoded in ν . Every spinstructure in the sense of Definition 7.5 defines a spin structure in the sense of [DK15] and viceversa. Note that in Lemma IV.26 in [DK15] it is shown that the datum of a spin structurecoincides with the more standard notion of a spin structure on S \ ( M ∩ S ◦ ) in the sense of areduction of the structure group of the tangent bundle to the connected two-fold covering ofGL + (2 , R ) ⊂ GL(2 , R ). Remark 7.7.
The datum of a spin structure ˜ I is equivalent to the datum of a lift ˜ I ∗ : Exit(Γ) → ∆ × Λ of ˜ I ∗ : Exit(Γ) → N ⊂ Set satisfying ν ∗ : ˜ I ∗ ≃ = ⇒ I ∗ . We will therefore refer to thedatum of ˜ I ∗ also as a spin structures on (Γ , V ). Definition 7.8.
An equivalence of two ribbon graphs with singularities (Γ , V ) ≃ (Γ ′ , V ′ ) con-sists of an equivalence of posets φ : Exit(Γ) ≃ −→ Exit(Γ ′ , V ′ ) which restricts to a bijection between V and V ′ .Let φ : Exit(Γ) ≃ −→ Exit(Γ ′ , V ′ ) be an equivalence of ribbon graphs with singularities. It isapparent that φ extends to a natural equivalence η φ : I ∗ Γ ′ ◦ φ ≃ == ⇒ I ∗ Γ between the relative incidence diagrams. Definition 7.9.
An equivalence between two ribbon graphs with singularities and spin structure(Γ , V, ˜ I ) , (Γ ′ , V ′ , ˜ I ′ ) consists of an equivalence φ : (Γ , V ) ≃ −→ (Γ ′ , V ′ ) of ribbon graphs withsingularities together with a lift η φ : ˜ I ∗ , ′ ◦ φ ≃ == ⇒ ˜ I ∗ , of ˜ I ∗ Γ ≃ = ⇒ I ∗ Γ η φ == ⇒ I ∗ Γ ′ ≃ = ⇒ ˜ I ∗ Γ . he main result of this section is the following. Theorem 7.10.
Let S be an oriented marked surface equipped with an ideal triangulation T andlet Σ = S \ ( M ∩ S ◦ ) be the open surface.1. For every spin structure U on Σ there exists a (Γ T , V T ) -parametrized perverse schober F U T . If U, U ′ are two equivalent spin structures on Σ there exists exists an equivalence ofparametrized perverse schobers F U T ≃ F U ′ T .
2. Given a parametrized perverse schober F T as constructed in Theorem 6.2, there exists aspin structure U on Σ and an equivalence of parametrized perverse schobers F U T ≃ F T .
3. Given two parametrized perverse schobers F T and F ′ T as constructed in Theorem 6.2 suchthat F T ≃ F U T and F ′ T ≃ F U ′ T with U, U ′ spin structures on Σ as considered in part 2. Thereexists an equivalence of spin structures U ≃ U ′ . Definition 7.11.
A ribbon graph with stops and singularities (Γ , V ) is called trivalent if allvertices have valency three.
Definition 7.12.
We define a subcategory M of ∆ × Λ spanned by the two objects { } × [2] , { } × [1] and morphisms generated under composition by• id { } × ( τ ) l : { } × [2] → { } × [2] for all l = 0 , . . . , id { } × ( τ ) l : { } × [1] → { } × [1] for all l = 0 , . . . , (cid:1) × δ l : { } × [1] → { } × [2] for 1 ≤ l ≤ Remark 7.13.
Let (Γ , V ) be a trivalent ribbon graph with singularities and I ∗ : Exit(Γ) → ∆ × Λ a spin structure. Then I ∗ factors through the inclusion M → ∆ × Λ . Proof of Theorem 7.10.
Consider the morphism of dg-algebras ϕ : k [ t ] t →− t −−−−−→ k [ t ] and con-sider the pullback functor ϕ ∗ : dgMod( k [ t ]) → dgMod( k [ t ]) . We define a functor Q : M op2 → dgCat k by Q ( { } × [2]) = D , Q ( { } × [1]) = k [ t ] , Q ( τ ) = T D , Q ( τ ) = ϕ ∗ , Q ( δ ) = i + x , Q ( δ ) = i − z , Q ( δ ) = i + y , where T D : D → D is the dg-functor from Lemma 5.14. Let T be an ideal triangulationand (Γ T , V T ) the dual ribbon graph with singularities. Given a spin structure U on (Γ T , V T )described in terms of ˜ I ∗ : Exit(Γ T ) → M we obtain a parametrized perverse schober F U T definedvia its left dual D L F U T : Entry(Γ T ) (˜ I ∗ ) op −−−−→ M op2 Q −→ dgCat k L −→ dgCat k [ W − ] → St . iven an equivalence of spin structures U ≃ U ′ , expressed in terms of an equivalence of ribbongraph with singularities and spin structure (Γ T , V T , ˜ I ∗ ) ≃ (Γ T , V T , ( ˜ I ∗ ) ′ ), it is immediate fromthe construction that there is an equivalence between F U T and F U ′ T . This shows statement 1.Let T be an ideal triangulation and consider a parametrized perverse schober F T as con-structed in Theorem 6.2. We now describe a spin structure U on (Γ T , V T ) such that F U T ≃ F T .We follow the iterative procedure and notation used in the proof of Theorem 6.2. Startingwith any ideal triangle T of T , one can directly find a spin structure U on (Γ T , V T ) suchthat F T ≃ F U T . We now continue by extending the spin structure in each induction step bygluing. Consider an ideal triangulation T n +1 , obtained from gluing two ideal triangulations T n and T equipped with spin structures. There is a first spin structure ˜ I ∗ ,n +12 : Exit(Γ T n +1 ) → M on (Γ T n +1 , V T n ) with the property that the restriction to Exit(Γ T n ) and Exit(Γ T ) recovers therespective spin structures of (Γ T n , V T n ) and (Γ T , V T ). At each edge e , used in the gluing of T n and T , connecting two vertices v n and v in Γ T n and Γ T , respectively, the corresponding spinstructure I ∗ ,n +12 on Γ T n +1 is locally of the form[2] = ˜ I ∗ ( v ) δ i ( τ ) l ←−−−−− [1] = ˜ I ∗ ( e ) δ j ( τ ) l −−−−−→ [2] = ˜ I ∗ ( v n ) . We obtain a further spin structure U on Γ T n +1 on by changing the spin structure I ∗ ,n +12 at eachedge e used in the gluing in the local picture to[2] = ˜ I ∗ ( v ) δ i ( τ ) l ←−−−−−−− [1] = ˜ I ∗ ( e ) δ j ( τ ) l −−−−−→ [2] = ˜ I ∗ ( v n ) , if the superscripts of the associated functors i e = i ± e and i ne = i ± e match. The resultingparametrized perverse schober F U T n +1 is then equivalent to F T n +1 . This shows statement 2.To show statement 3, we describe the spin structure constructed in the above proof ofstatement 2. Consider an ideal triangulation T of S and the associated spin structure ˜ I ∗ :Exit(Γ T ) → M ⊂ Λ . Using the paracyclic equivalence τ : [2] → [2] sufficiently often, wesee that locally at the circle around each marked point in S , the spin structure on Γ T is up tonatural equivalence of the following form. [1][1] [2] [1][1] [2] [2] [1][1] . . . . . . δ δ δ δ δ δ δ By a classical result, spin structures on Σ are classified by H (Σ , Z ). This means that the spinstructure ˜ I ∗ is fully determined by the circle around each marked point. It follows that any twospin structures as constructed in the proof of statement 2 are equivalent. Let T be an ideal triangulation of a surface and consider the construction of the parametrizedperverse schober F T of Theorem 6.2. Given any stable ∞ -category D , we can construct aparametrized perverse schober F T ( D ) by replacing the spherical adjunction at each vertex withthe spherical adjunction f ∗ : D ←→ Fun( S , D ) : f ∗ nd the autoequivalence T (used there for fixing the correct signs) by the two-fold suspension ofthe cotwist functor T Fun( S , RMod R ) [2] of the adjunction f ∗ ⊣ f ∗ . The goal of this sections is todiscuss which results of this paper translate from the case D = D ( k ) to more general coefficients.We begin by noting that the main result which works irrespective of the choice of D is theconstruction of the derived equivalence µ e : H (Γ , F T ( D )) ≃ H (Γ , F T ′ ( D ))of Section 6.4 associated to the flip of the edge e of the triangulation T .More results extend in the case where D = RMod R is the ∞ -category of right modules ofan E ∞ -ring spectrum R . We can for example choose R to be the sphere spectrum, so that D ≃ Sp is the ∞ -category of spectra. The construction of the spherical/exceptional objects S e , the objects C p and the projective objects P e given in Section 6.3 works in the same way for H (Γ T , F T (RMod R )). Denote by R [ t n ] the free algebra object of RMod R generated by R [ n ]. Notethat if R = k is a commutative ring, there exists an equivalence R [ t n ] ≃ k [ t n ]. We show below inProposition 7.16 that there exists an equivalence of ∞ -categories Fun( S , RMod R ) ≃ RMod R [ t ] .We thus find a compact generator of the ∞ -category H (Γ T , F T (RMod R )), as in Proposition 6.8,given by the sum of the images of R [ t ] ∈ RMod R [ t ] ≃ Fun( S , RMod R ) under the left adjointsof the evaluation functors to the edges of Γ T . Contrary to the dg-setting, it is not clear if theendomorphism algebra of the compact generator admits an explicit description.The conceptual reason for a relation between the construction of F T ( D ( k )) and spin struc-tures on the open surface Σ in Section 7.1 is the observation that the suspended cotwist functor T Fun( S , D ( k )) [2] is an involution, i.e. (cid:0) T Fun( S , D ( k )) [2] (cid:1) ≃ id Fun( S , D ( k )) . It seems likely that if the cotwist functor T Fun( S , D ) [2] is also an involution Theorem 7.10generalizes to F T ( D ). The remainder of this section consists of a proof of Proposition 7.16 anda conjecture for a description of T Fun( S , D ) and an algebraic description of T Fun( S , RMod R ) .The following lemma is a generalization of Lemma 5.3. Lemma 7.14.
Consider the morphism of simplicial sets g : L → ∗ and the associated pullbackfunctor g ∗ : RMod R → Fun( L, RMod R ) . There exists an equivalence of ∞ -categories R -linear ∞ -categories Fun( L, RMod R ) ≃ RMod R [ t ] such that the following diagram commutes. RMod R Fun( L, RMod R ) RMod R [ t ] g ∗ φ ∗ ≃ (81) Here φ ∗ denotes the pullback functor along the morphism of R -algebras R [ t ] → R , determinedon the generator by the morphism R id −→ R in RMod R .Proof. We observe that Fun( L, RMod R ) admits a compact generator X , given by the diagram R [ t ] · t −−→ R [ t ] in RMod R . It follows that Fun( L, RMod R ) lies in the essential image of thefunctor θ : Alg(RMod R ) → LinCat R , introduced in Section 2.2. Using the involved universalproperties, we determine a morphism of ring spectra µ : R [ t ] → End R ( X ) by mapping t tothe diagram R [ t ] R [ t ] R [ t ] R [ t ] · t · t · t · t epresenting an endomorphism of X . We wish to show that µ is an equivalence. Given two R -linear ∞ -categories C , D , we denote by Lin R ( C , D ) the ∞ -category of R -linear functors from C to D . We denote by R Lin( C , D ) the ∞ -category of functors from C to D that admit an R -linearleft adjoint. Denote by S the ∞ -category of spaces and consider the functorsMap Alg(RMod R ) (cid:0) End R ( X ) , - (cid:1) , Map
Alg(RMod R ) (cid:0) R [ t ] , - (cid:1) : Alg(RMod R ) → S corepresented by End R ( X ) and R [ t ], respectively. Composition with µ induces a natural trans-formation µ ∗ : Map Alg(RMod R ) (cid:0) End R ( X ) , - (cid:1) −→ Map
Alg(RMod R ) (cid:0) R [ t ] , - (cid:1) . Note that µ is an equivalence if and only if µ ∗ is pointwise an equivalence of spaces. Let B ∈ Alg(RMod R ). It follows from [Lur17, 4.8.5.6,4.8.4.1] that there exists an equivalence ofspacesMap Alg(RMod R ) (cid:0) End R ( X ) , B (cid:1) ≃ Lin R (Fun( L, RMod R ) , RMod B ) × Fun( { X } , RMod B ) { B } . Consider the following equivalences of functors in Fun(LinCat R , Cat ∞ ).Lin R (Fun( L, RMod R ) , -) ≃ R Lin((-) op , Fun( L, RMod R )) ≃ Fun( L, R Lin((-) op , RMod R )) ≃ Fun( L, Lin R (RMod R , -)) . The equivalence Lin R (Fun( L, RMod R ) , -) ≃ Fun( L, Lin R (RMod R , -)) restricts to an equivalenceof spacesLin R (Fun( L, RMod R ) , RMod B ) × Fun( { X } , RMod B ) { B } ≃ Fun( L, RMod B ) × Fun( L , RMod B ) { B } . We further note the equivalences of spacesFun( L, RMod B ) × Fun( L , RMod B ) { B } ≃ Map
RMod B ( B, B ) ≃ Map
RMod R ( R, B )and Map
RMod R ( R, B ) ≃ Map
Alg(RMod R ) (cid:0) R [ t ] , B (cid:1) . Tracing through the above equivalences, we see that µ ∗ ( B ) is an equivalence of spaces, showingthat µ ∗ is a natural equivalence. We deduce that End R ( X ) and R are equivalent in Alg(RMod R ).This further implies that there exists an equivalence of R -linear ∞ -categories Fun( L, RMod R ) ≃ RMod R [ t ] .The R -linear functors φ ∗ , g ∗ : RMod R → RMod R [ t ] are fully determined by φ ∗ ( R ) respec-tively g ∗ ( R ), see [Lur17, Section 4.8.4]. The R [ t ]-module φ ∗ ( R ) corresponds to the diagram R id −→ R in RMod R , as does the R [ t ]-module g ∗ ( R ). It follows that diagram (81) commutes. Lemma 7.15.
1. There exists a pushout diagram in
LinCat R as follows. Fun( L, RMod R ) RMod R RMod R Fun( S , RMod R ) g ! g ! p i ! i ! (82)
2. Let n ≥ . There exists a pushout diagram in LinCat R as follows. Fun( S n − , RMod R ) RMod R RMod R Fun( S n , RMod R ) f ! f ! p i ! i ! (83) roof. The proof of Lemma 5.4 directly generalizes.We now prove the analogue of Proposition 5.5.
Proposition 7.16.
Let n ≥ . There exists an equivalence of R -linear ∞ -categories Fun( S n , RMod R ) ≃ RMod R [ t n − ] , such that the following diagram in LinCat R commutes. Fun( S n , RMod R ) RMod R [ t n − ] RMod R Fun( S n , RMod R ) RMod R [ t n − ] ≃ i ∗ Gf ∗ φ ∗ ≃ (84) Here G denoted the monadic functor and φ ∗ the pullback functor along the morphism of R -algebras φ : R [ t n − ] → R determined on the generator by the morphism R [ n − −→ R in RMod R .Proof. Consider the following biCartesian square in RMod R . R [ n −
1] 00 R [ n ] (cid:3) Applying the colimit preserving free R -algebra functor RMod R → Alg(RMod R ) yields the fol-lowing pushout diagram of R -algebras. R [ t n − ] RR R [ t n ] t n − t n − p (85)Consider the morphism of ring spectra R [ t ] t t +1 −−−−−−→ R [ t ], determined from the universalproperty by the morphism R t −−−−−→ R [ t ] in RMod R . Using the commutativity of the diagram R [ t ] RR [ T ] t t t +1 t it follows that for n = 1 the image of the diagram (85) under θ : Alg(RMod) → LinCat R isequivalent to the pushout diagram in (82). It follows that there exists an equivalence of R -linear ∞ -categories Fun( S , RMod R ) ≃ RMod R [ t ] . Using that the monadic functor G is equivalentto the pullback along R → R [ t ], we obtain that the upper half of the diagram (36) commutes.Using that f ! ◦ i ! ≃ id RMod R , see Remark 5.6, we obtain the following commutative diagram.Fun( L, RMod R ) RMod R RMod R Fun( S , RMod R ) RMod R p g ! g ! i ! id i ! id f ! (86) he diagram (86) is equivalent to the image under θ of the following diagram in Alg(RMod R ). R [ t ] RR R [ t ] R p idid (87)By the universal property of the pushout in Alg(RMod R ) there exists a unique morphism ofring spectra R [ t ] → R such that (87) commutes. Such a map is given by φ . It follows that thefunctor f ! is equivalent to θ ( φ ) and using [Lur17, 4.6.2.17] also that the functor f ∗ is equivalentto the pullback functor along φ .For n ≥
2, we can continue by induction and as before. The image of (85) under the functor θ is the pushout diagram in (83). We thus find the desired equivalence Fun( S n , RMod R ) ≃ RMod R [ t n − ] so that the upper half of diagram (84) commutes. Analogous to the case n = 1, itcan be checked that the lower half of the diagram (84) commutes.Our proof of Proposition 5.7 characterizing the cotwist functor of the spherical adjunction f ∗ ⊣ f ∗ does not directly generalize to the spectral setting. We conjecture the following. Conjecture 7.17.
1. Let n ≥ and let ϕ : R [ t n − ] → R [ t n − ] be the equivalence of ring spectra determined by ǫ ( t n − ) = − t n − . There exists a commutative diagram in LinCat R as follows. Fun( S n , RMod R ) Fun( S n , RMod R )RMod R [ t n − ] RMod R [ t n − ] T Fun(
Sn,
RMod R ) ≃ ≃ ϕ ∗ [ − n ]
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Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Ham-burg, Germany
Email address: [email protected]@uni-hamburg.de