2.1 Grading of surfaces and curves

We recall the definitions of grading for surfaces and curves on them, following [Reference Haiden, Katzarkov and KontsevichHKK17]. This is the lowest-dimensional case of the more general notions of grading for symplectic manifolds and their Lagrangian submanifolds [Reference SeidelSei00].

For us, grading structures arise as horizontal foliations of quadratic differentials. This foliation, $\mathrm{hor}(\varphi)$ , is defined on the complement of the zeros and poles of a quadratic differential $\varphi$ on a Riemann surface $\mathbf{S}$ by

\[\mathrm{hor}(\varphi)(p):=\{v\in T_p{\mathbf{S}}\mid \varphi(v,v)\in\mathbb R_{\geqslant 0}\}\in\mathbb P( T_p {\mathbf{S}}). \]

If $({\mathbf{S}},\nu)$ is a graded surface, then any immersed loop $c\colon S^1\to \mathbf{S}$ has a Maslov index, $\mathrm{ind}_\nu(c)$ , which records the number of times (more precisely an intersection number) that $\nu$ becomes tangent to c. The sign is normalized so that if c is a counterclockwise loop around the origin in $\mathbb{C}$ and $\nu$ is the horizontal foliation of the quadratic differential $z^kdz^2$ , then $\mathrm{ind}_\nu(c)=k+2$ . If $\nu$ is defined in a punctured neighbourhood of a point $p\in\mathbf{S}$ , then define $\mathrm{ind}_\nu(p):= \mathrm{ind}_\nu(c)$ , where c is a small counterclockwise loop around p.

We note that if I is an interval, then there is a $\mathbb Z$ -torsor of choices of $\tilde{c}$ for given c. If $I=S^1$ , then $\tilde{c}$ exists if and only if $\mathrm{ind}_\nu(c)=0$ . To a pair of graded curves $(I_1,c_1,\tilde{c}_1)$ , $(I_2,c_2,\tilde{c}_2)$ intersecting transversely in a point $p=c_1(t_1)=c_2(t_2)$ one can assign an integer $i_p(c_1,c_2)$ as the class in $\pi_1(\mathbb P(T_p{\mathbf{S}}))=\mathbb Z$ given by $\tilde{c}_1(t_1)\cdot\kappa\cdot\tilde{c}_2(t_2)^{-1}$ , where $\kappa$ going from $\dot{c}_1(t_1)$ to $\dot{c}_2(t_2)$ is given by counterclockwise rotation in $T_p\mathbf{S}$ by an angle less than $\pi$ .

We will also require a definition of the intersection index $i_p(c_1,c_2)$ in the case where two graded curves $(I_1,c_1,\tilde{c}_1)$ , $(I_2,c_2,\tilde{c}_2)$ intersect at a common endpoint $p\in\mathbf{S}$ at which $\nu$ is undefined, but rather defined on a small punctured disc centred at p. To do this, choose a small arc $\alpha\colon[0,1]\to \mathbf{S}$ starting on $c_1$ and ending on $c_2$ and going counterclockwise around p and choose an arbitrary grading of $\alpha$ . Then $i_p(c_1,c_2):= i_{\alpha(0)}(c_1,\alpha)-i_{\alpha(1)}(c_2,\alpha)$ is independent of the choice of the graded curve $\alpha$ (cf. [Reference Haiden, Katzarkov and KontsevichHKK17, equation (3.17)]).

2.2 Weighted marked surfaces and mixed angulations

A mixed angulation is a collection of polygons with possibly varying numbers of edges and an identification of certain pairs of edges. We will consider sets of mixed angulations, related by a sequence of flips, on the same underlying surface. To set this up consistently, we need a notion of marked surface.

For simplicity, we also assume that $\mathbf{S}$ is connected. An isomorphism of weighted marked surfaces $\mathbf{S}_{\mathbf{w}}\to\mathbf{S}_{\mathbf{w}}'$ is an isomorphism of graded surfaces $(f,h)\colon ({\mathbf{S}},\nu)\to (\mathbf{S}',\nu')$ with $f(\mathbf{M})=\mathbf{M}'$ , $f(\Delta)=\Delta'$ , and $\mathbf{w}'\circ f=\mathbf{w}$ .

We define $d(x):= \mathbf{w}(x)+2$ and refer to this as the degree of $x\in\Delta$ .

The notion of a mixed-angulated surface is dual to that of an S-graph in the sense of [Reference Haiden, Katzarkov and KontsevichHKK17].

Given the data of $d(h_1,h_2)$ for consecutive half-edges, we define $d(h_1,h_2)$ for other pairs $h_1$ , $h_2$ of half-edges attached to the same vertex v so that $d(h_1,h_2)=0$ if $h_1=h_2$ and $d(h_1,h_3)=d(h_1,h_2)+d(h_2,h_3)$ if the sequence $(h_1,h_2,h_3)$ is compatible with the cyclic/total order at v. If v is an internal vertex, then this determines $d(h_1,h_2)$ for all pairs $(h_1,h_2)$ , and if v is a boundary vertex then this determines $d(h_1,h_2)$ for all pairs with $h_1\leqslant h_2$ .

Figure 2.

Example of mixed-angulated surface and dual S-graph. Numbers $d(h_1,h_2)$ for consecutive half-edges $h_1$ , $h_2$ are shown.

The edges of the dual S-graph have a canonical grading such that $i(X,Y)=0$ if $X\in\mathbb A$ and Y is the dual edge of the S-graph. Then $d(h_1,h_2)$ is just the intersection index of the two corresponding edges of the S-graph at the given vertex.

Next, we introduce the flips of mixed angulations. We say an arc in a mixed angulation is a monogon arc if it encloses a monogon.

Figure 3.

The forward flip at a usual arc.

Figure 4.

The forward flip at a monogon arc.

The exchange graph $\operatorname{EG}(\mathbf{S}_{\mathbf{w}})$ of the weighted marked surface $\mathbf{S}_{\mathbf{w}}$ is the directed graph whose vertices are mixed angulations and whose oriented edges are forward flips between them.

We will refer to the two cases of flips, as in Figures 3 and 4, as the usual and the monogon case, respectively.

Remark 2.10. The forward flip operation on an S-graph

\[ \mathbb{S}=\mathbb{A}^*\to\mathbb{S}^\sharp_{e}=(\mathbb{A}^\sharp_\gamma)^* \]

can concretely be described as follows, where e in $\mathbb{S}$ is the dual edge of $\gamma$ in $\mathbb{A}$ . The arc e remains in $\mathbb{S}^\sharp_{e}$ but its grading will be shifted downward by one. Moreover, in the usual case, the weights of both endpoints of e are bigger than 1; in the monogon case, the degree of exactly one endpoint of e is 1. For $h\neq{e}\in\mathbb{S}$ , we flip it to h’ as follows.

• – In the usual case, if there is an oriented intersection x of index 1 from e to h at a singular point Z, then move the endpoint Z of h along e with respect to Z. See Figure 3 for $h=a,b$ .

• – In the monogon case, if there is an oriented intersection x of index 1 from e to $\alpha$ at a singular point Z, then apply the square of the braid twist $B_{e}$ along e (cf. [Reference King and QiuKQ20, Definition 3.10]) as shown in Figure 4 to h and then $h'=B_{e}^2(h)$ .

The exchange graph $\operatorname{EG}_S(\mathbf{S}_{\mathbf{w}})$ of S-graphs of $\mathbf{S}_{\mathbf{w}}$ is the directed graph whose vertices are S-graphs and whose oriented edges are forward flips between them.

2.3 Quadratic differentials

Let C be a compact Riemann surface and $\varphi$ a quadratic differential which is holomorphic and non-vanishing away from a finite subset $D\subset C$ . Each point in D should be either a zero, a pole, a regular point, or a transcendental singularity of $\varphi$ of the form $\exp(f(z))g(z)dz^2$ where f and g are meromorphic and f has a pole of order n at $z=0$ , called an exponential singularity of order n. Thus, D consists of the zeros, poles, and exponential singularities of $\varphi$ , as well as possibly some additional marked points where $\varphi$ is holomorphic and non-vanishing.

Besides the complex-analytic point of view on quadratic differentials, there is also a metric point of view. Indeed, $|\varphi|$ is a flat Riemannian metric on $C\smallsetminus D$ , that is, $C\smallsetminus D$ has the structure of a metric space which is locally isometric to the Euclidean plane. However, $C\smallsetminus D$ is in general not complete as a metric space and the metric completion, $\overline{C\smallsetminus D}$ , adds a finite number of conical points. More precisely, $\overline{C\smallsetminus D}$ has a conical point with cone angle $(n+2)\pi$ for each zero of order n of $\varphi$ , a conical point of cone angle $\pi$ for each simple pole of $\varphi$ , and n infinite-angle conical points for each exponential singularity of order n, and a (non-singular) point with cone angle $2\pi$ for each regular point of $\varphi$ in D. The metric is already complete near higher-order poles of $\varphi$ , so these do not lead to any additional conical points. Proofs of these claims may be found in [Reference Haiden, Katzarkov and KontsevichHKK17].

A flat surface has a total area which may be finite or infinite. In fact, the area of $\varphi$ is finite if and only if $\varphi$ has no higher-order poles and no exponential singularities. The finite-area case is interesting from the point of view of ergodic theory and has been much studied. Quadratic differentials $\varphi$ with infinite area, on the other hand, are closely related to mixed angulations. This was already noted in [Reference Haiden, Katzarkov and KontsevichHKK17] in terms of the dual language of S-graphs. Let us recall this relation. We need to assume also that $|\varphi|$ has at least one conical point, that is, exclude the flat plane and flat cylinder. For generic choice of $\theta\in\mathbb R$ , the horizontal foliation of $e^{i\theta}\varphi$ then has the structure of a horizontal strip decomposition. This means the following. After possibly replacing $\varphi$ by $e^{i\theta}\varphi$ , we can assume that the horizontal foliation of $\varphi$ , $\mathrm{hor}(\varphi)$ , has no finite-length leaves. Then there are two types of leaves, or trajectories.

1. (i°) Separating trajectories. Approach a conical point in one direction and escape ‘to infinity’ (towards a higher-order pole or exponential singularity) in the other direction.

2. (ii°) Generic trajectories. Avoid the conical points and go to infinity in both directions.

Each separating trajectory belongs to some conical point, by definition, and a conical point with cone angle $n\pi$ gives rise to n separating trajectories. These separating trajectories cut the surface into horizontal strips, foliated by one-parameter families of generic trajectories. Horizontal strips are isometric to $(0,a)\times\mathbb R$ for some $a\in (0,\infty]$ , their height. Horizontal strips of finite (respectively, infinite) height have four (respectively, two) separating trajectories on their boundary, some of which are possibly identified. Choosing a generic trajectory from each horizontal strip gives a mixed angulation whose edges are these horizontal trajectories. Each conical point of cone angle $n\pi$ corresponds to an n-gon (see Figure 5). Edges corresponding to infinite-height horizontal strips live on the boundary of the mixed angulation, that is, belong to just one polygon.

Figure 5.

A 5-gon around a triple zero of a quadratic differential. Parts of the horizontal foliation are shown by thin curved trajectories, the mixed angulation by the pentagon connecting the five bullet marked points, and the dual S-graph by the radial spokes through the central circled point.

The dual S-graph of the mixed angulation is obtained as the union of the finite-length geodesics which connect the conical points on opposite boundaries of the strip (which can be two different points or the same conical point). The quadratic differential is then entirely encoded in terms of the S-graph and a positive number Z(e) in the upper half-plane for each edge, which is the vector between the conical points on the boundary of the horizontal strip e.

One issue with the above construction of a mixed angulation from a quadratic differential is that the mixed angulation lives on the compact surface C which we cannot take to be the weighted marked surface $\mathbf{S}_{\mathbf{w}}$ in the case where $\varphi$ has poles of order greater than 2 or exponential singularities. To correct this, we define $\mathbf{S}_{\mathbf{w}}(C,D,\varphi)=({\mathbf{S}},\mathbf{M},\Delta,\mathbf{w},\nu)$ associated with $(C,D,\varphi)$ as follows:

• – $\mathbf{S}$ is the oriented real blow-up of C in the poles of order at least 3 and exponential singularities;

• – $\mathbf{M}\cap\mathrm{int}(\mathbf{S})$ is the set of second-order poles;

• – $\mathbf{M}\cap \partial \mathbf{S}$ is the set of directions from which horizontal trajectories approach (there are n such directions for a pole of order $n+2$ and n different $\mathbb Z$ -torsors’ worth for an exponential singularity of order n);

• – $\Delta$ is the set of conical points (simple poles, marked points, zeros);

• – $(\mathbf{w}(x)+2)\pi=d(x)\pi$ is the cone angle of the conical point;

• – $\nu$ is the horizontal foliation of $\varphi$ .

We consider certain moduli spaces of quadratic differentials which give rise to a given weighted marked surface. These are special cases of the moduli spaces $\mathcal M(S)$ considered in [Reference Haiden, Katzarkov and KontsevichHKK17], as here we restrict to flat surfaces of infinite area.

There is a well-known affine structure on $\operatorname{FQuad}^{}(\mathbf{S}_{\mathbf{w}})$ given by the period map. This is defined as follows. First, let $\Sigma\to {\mathbf{S}}\smallsetminus (\mathbf{M}\cup \Delta)$ be the double cover whose fibre over p is the two-element set of orientations of the line $\nu(p)\subset T_p\mathbf{S}$ . (If $\nu$ is the horizontal foliation of a quadratic differential $\varphi$ , then this is just the covering where $\sqrt{\varphi}$ becomes well defined.) Let

\[\operatorname{H}_1(\mathbf{S}_{\mathbf{w}}):= H_1({\mathbf{S}}\smallsetminus \mathbf{M},\Delta;\mathbb Z\times_{\mathbb Z/2}\Sigma) \]

where $\mathbb Z\times_{\mathbb Z/2}\Sigma$ is the local system of abelian groups whose fibre over p is the abelian group generated by $\Sigma_p=\{o_1,o_2\}$ with relation $o_1=-o_2$ , thus non-canonically isomorphic to $\mathbb Z$ . Note that an isomorphism of weighted marked surfaces $\mathbf{S}_{\mathbf{w}}\to \mathbf{S}_{\mathbf{w}}'$ induces an isomorphism of abelian groups $\operatorname{H}_1(\mathbf{S}_{\mathbf{w}})\to \operatorname{H}_1(\mathbf{S}_{\mathbf{w}}')$ . Given a triple $(C,D,\phi)$ , there is a map

(2.1) \begin{equation} \int\colon\operatorname{H}_1(\mathbf{S}_{\mathbf{w}}(C,D,\phi))\to \mathbb C,\quad \gamma\mapsto \int_{\gamma}\sqrt{\phi}\end{equation}

so if $(C,D,\phi,f)\in\operatorname{FQuad}^{}(\mathbf{S}_{\mathbf{w}})$ then we get a map $\operatorname{H}_1(\mathbf{S}_{\mathbf{w}})\to\mathbb C$ . These combine to give a continuous map

(2.2) \begin{gather} \Pi\colon\operatorname{FQuad}^{}(\mathbf{S}_{\mathbf{w}})\longrightarrow\mathrm{Hom}(\operatorname{H}_1(\mathbf{S}_{\mathbf{w}}),\mathbb C)\end{gather}

which turns out to be a local homeomorphism (see [Reference Haiden, Katzarkov and KontsevichHKK17, Theorem 2.1], which also includes the case of exponential singularities). The above map may be used to define an affine structure on $\operatorname{FQuad}^{}(\mathbf{S}_{\mathbf{w}})$ by pulling back the one on $\mathrm{Hom}(\operatorname{H}_1(\mathbf{S}_{\mathbf{w}}),\mathbb C)\cong \mathbb C^k$ .

3.1 Preliminaries

We first recall some basic definitions around subcategories of additive/abelian/triangulated categories.

In an additive category $\operatorname{\mathcal{C}}$ with a subcategory $\mathcal{B}$ , we denote

\[ \mathcal{B}^{\perp_{\operatorname{\mathcal{C}}}}:= \{C\in\operatorname{\mathcal{C}}:\operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,C)=0\ \forall B\in \mathcal{B}\}. \]

We may write $\mathcal{B}^{\perp}$ when $\operatorname{\mathcal{C}}$ is implied. The subcategory ${}^{\perp_{\operatorname{\mathcal{C}}}}\mathcal{B}$ is defined similarly.

For subcategories $\mathcal{T},\mathcal{F}$ of an abelian category $\operatorname{\mathcal{H}}$ , denote

$$ \mathcal{T}*\mathcal{F}:= \{M\in\operatorname{\mathcal{H}}\mid\text{$\exists$ s.e.s. $0\to T\to M\to F\to0$ s.t. $T\in\mathcal{T}, F\in\mathcal{F}$} \},$$

where s.e.s. is short exact sequence. If additionally $\operatorname{Hom}(\mathcal{T},\mathcal{F})=0$ , then we write $\mathcal{T}\perp\mathcal{F}$ for $\mathcal{T}*\mathcal{F}$ . For a subcategory $\operatorname{\mathcal{C}}$ of $\operatorname{\mathcal{H}}$ , denote by $\langle\operatorname{\mathcal{C}}\rangle$ the full subcategory of $\operatorname{\mathcal{H}}$ consisting of objects that admit a filtration with factors in $\operatorname{\mathcal{C}}$ .

A map $f\colon E\to F$ in an arbitrary additive category is called right minimal if it does not have a direct summand of the form $E'\to0$ . Similarly for left minimal.

Let $\operatorname{\mathcal{C}}$ be an additive category with a full subcategory $\mathcal{X}$ . A right $\mathcal{X}$ -approximation of an object C in $\operatorname{\mathcal{C}}$ is a morphism $f\colon X\to C$ with $X\in\mathcal{X}$ , such that $\operatorname{Hom}(X',f)$ is an epimorphism for any $X'\in \mathcal{X}$ . Dually, a left $\mathcal{X}$ -approximation of an object C in $\operatorname{\mathcal{C}}$ is a morphism $g\colon C\to X$ with $X\in\mathcal{X}$ , such that $\operatorname{Hom}(g,X')$ is an epimorphism for any $X'\in\mathcal{X}$ . A minimal right/left $\mathcal{X}$ -approximation is a right $\mathcal{X}$ -approximation that is also right/left minimal. We say $\mathcal{X}$ is contravariantly finite (respectively, covariantly finite) if every object in $\operatorname{\mathcal{C}}$ admits a right (respectively, left) $\mathcal{X}$ -approximation. It is functorially finite if it is both contravariantly finite and covariantly finite.

A torsion pair in an abelian category $\operatorname{\mathcal{H}}$ consists of two additive subcategories $\mathcal{T}$ and $\mathcal{F}$ such that $\operatorname{\mathcal{H}}=\mathcal{T}\perp\mathcal{F}$ . Denote by $\operatorname{Sim}\operatorname{\mathcal{H}}$ the set of simples of $\operatorname{\mathcal{H}}$ . An abelian category $\operatorname{\mathcal{H}}$ is called a length category if any object in $\operatorname{\mathcal{H}}$ admits a finite filtration with factors in the set of simples $\operatorname{Sim}\operatorname{\mathcal{H}}$ . The abelian category $\operatorname{\mathcal{H}}$ is called $\textit{finite}$ if it is a length category with a finite number of simples.

Similarly, we have the corresponding notions and notation in a triangulated category $\operatorname{\mathcal{D}}$ , replacing s.e.s. by triangles. A t-structure $\mathcal{P}$ of $\operatorname{\mathcal{D}}$ is the torsion part of some torsion pair of $\operatorname{\mathcal{D}}$ which is closed under shift (i.e. $\mathcal{P}[1]\subset\mathcal{P}$ ). We will write it as $\operatorname{\mathcal{D}}=\mathcal{P}\perp\mathcal{P}^{\perp}$ . Such a t-structure is bounded if for any object M in $\operatorname{\mathcal{D}}$ we have $M[\gg0]\in\mathcal{P}$ and $M[\ll0]\in\mathcal{P}^\perp$ .

The heart of a t-structure $\mathcal{P}$ is $\operatorname{\mathcal{H}}=\mathcal{P}^\perp[1]\cap\mathcal{P}$ . An additive subcategory $\operatorname{\mathcal{H}}$ of a triangulated category $\operatorname{\mathcal{D}}$ is the heart of a bounded t-structure of $\operatorname{\mathcal{D}}$ if and only if the following two conditions hold (see [Reference BridgelandBri07, Lemma 3.2]):

• – $\operatorname{Hom}(\operatorname{\mathcal{H}}[a],\operatorname{\mathcal{H}}[b])=0$ for any $a\gt b$ in $\mathbb{Z}$ ;

• – any object E in $\operatorname{\mathcal{D}}$ admits a finite filtration whose associated graded has in each degree an object of the form H[k] with $H\in\operatorname{\mathcal{H}}$ and $k\in \mathbb{Z}$ .

The additive subcategories of a triangulated category $\operatorname{\mathcal{D}}$ satisfying the above two conditions will also be referred to as the hearts of $\operatorname{\mathcal{D}}$ .

There is a natural partial order on hearts corresponding to the reverse inclusion relation on the associated t-structures, that is, $\operatorname{\mathcal{H}}_1\leqslant \operatorname{\mathcal{H}}_2 \Leftrightarrow \mathcal{P}_1\supset\mathcal{P}_2$ .

3.2 Tilting theory

In this section we discuss the tilting of hearts, and in particular the tilting with respect to simple objects which are not necessarily rigid.

Recall simple Happel–Reiten–Smalø tilting.

A simple S is rigid if $\operatorname{Ext}^1(S,S)=0$ . When $S\in\operatorname{Sim}\operatorname{\mathcal{H}}$ is rigid, the simple tilting with respect to S is particularly easy to determine (see Remark 3.3). Next, we consider the simple tilting with respect to a possibly non-rigid simple object. Such simple tiltings were also described in [Reference Koenig and YangKY14, Definition 7.5, Proposition 7.10].

Proposition 3.2. Let S be a simple in a heart $\operatorname{\mathcal{H}}\in\operatorname{EG}(\operatorname{\mathcal{D}})$ , which is a length category. Then $\langle S\rangle$ is functorially finite in $\operatorname{\mathcal{H}}$ . Furthermore, if both $\operatorname{\mathcal{H}}$ and $\operatorname{\mathcal{H}}^\sharp_S$ are finite, then

(3.1) \begin{gather} \operatorname{Sim}\operatorname{\mathcal{H}}^\sharp_S = \{S[1]\}\cup\{ \psi^\sharp_{S}(X)\mid X\in \operatorname{Sim}\operatorname{\mathcal{H}},X\neq S\},\end{gather}

where $\psi^\sharp_{S}(X)=\operatorname{Cone}(f)[-1]$ for f the left minimal $\langle S[1]\rangle$ -approximation of X. Similarly, if both $\operatorname{\mathcal{H}}$ and $\operatorname{\mathcal{H}}^\flat_S$ are finite, then

(3.2) \begin{gather} \operatorname{Sim}\operatorname{\mathcal{H}}^\flat_S = \{S[-1]\}\cup\{ \psi^\flat_{S}(X)\mid X\in \operatorname{Sim}\operatorname{\mathcal{H}},X\neq S\},\end{gather}

where $\psi^\flat_{S}(X)=\operatorname{Cone}(g)$ for g the right minimal $\langle S[-1]\rangle$ -approximation of X.

The statement that $\langle S\rangle$ is functorially finite can also be found in [Reference DugasDug15, Theorem 3.3].

Proof of Proposition 3.2. Given $M\in \operatorname{\mathcal{H}}$ , we define

(3.3) \begin{gather} \operatorname{top}_S(M):= S\otimes\operatorname{Hom}(M,S)^*\quad\text{and}\quad \operatorname{sub}_S(M):= \ker( M\xrightarrow{\text{ev}} \operatorname{top}_S(M) )\end{gather}

and inductively define

\[ \begin{cases} \operatorname{sub}^m_S(M)=\operatorname{sub}_S(\operatorname{sub}^{m-1}_S(M)), \\[3pt] \operatorname{top}^m_S(M)=\operatorname{top}_S(\operatorname{sub}^{m-1}_S(M))\end{cases} \]

using (3.3) and set $\operatorname{sub}^0_S(M)=M$ . For any $M\neq0$ in $\operatorname{\mathcal{H}}$ , there exists $j\in\mathbb{Z}_{\gt0}$ with $\operatorname{top}^1_S(M)\ne0$ and $\operatorname{top}^{j+1}_S(M)=0$ , since $\operatorname{\mathcal{H}}$ is a length category. Thus, we obtain the following filtration of short exact sequences.

(3.4)

In particular, we have a short exact sequence

(3.5)

where $\operatorname{top}^\bullet_S(M)$ admits a filtration with factors $\operatorname{top}^i_S(M)$ in $\langle S\rangle$ for $1\leqslant i\leqslant j$ . Thus, $\operatorname{top}^\bullet_S(M)$ is also in $\langle S\rangle$ . As $\operatorname{top}^{j+1}_S(M)=\operatorname{top}_S (\operatorname{sub}^j_S(M))=0$ , we have $\operatorname{sub}^j_S(M)\in {}^{\perp_{\operatorname{\mathcal{H}}}} S$ . This implies there is a torsion pair $\operatorname{\mathcal{H}}={}^{\perp_{\operatorname{\mathcal{H}}}} S\perp \langle S\rangle$ . By applying $\operatorname{Hom}(-,S)$ to (3.5), we see that $f_0$ is the left $\langle S\rangle$ -approximation of M. Hence, $\langle S\rangle$ is covariantly finite. Dually, there is a torsion pair $\operatorname{\mathcal{H}}=\langle S\rangle\perp S^{\perp_{\operatorname{\mathcal{H}}}}$ and $\langle S\rangle$ is contravariantly finite. So $\langle S\rangle$ is functorially finite.

Next, suppose that $\operatorname{\mathcal{H}}_S^\sharp$ is also finite. Then, $\langle S[1]\rangle$ is functorially finite in $\operatorname{\mathcal{H}}_S^\sharp$ . Let $X\not\simeq S$ be any other simple in $\operatorname{\mathcal{H}}$ , so that $X\in {}^{\perp_{\operatorname{\mathcal{H}}}} S\subset\operatorname{\mathcal{H}}_S^\sharp$ . Let T be a simple subobject of X in $\operatorname{\mathcal{H}}_S^\sharp$ . As X is simple in $\operatorname{\mathcal{H}}$ and T is simple in $\operatorname{\mathcal{H}}_S^\sharp$ , there are short exact sequences

(3.6)

in $\operatorname{\mathcal{H}}$ and $\operatorname{\mathcal{H}}_S^\sharp$ , respectively. Thus $M'=M[1]$ . As T is a simple in $\operatorname{\mathcal{H}}^\sharp_S$ , we have $\operatorname{Hom}(T,S[1])=0$ . Applying $\operatorname{Hom}(-,S[1])$ to the second short exact sequence in (3.6), we see that f is the left $\langle S[1]\rangle$ -approximation of X. It must be minimal, since f is an epimorphism in $\operatorname{\mathcal{H}}$ . Thus, $T=\psi^\sharp_{S}(X)$ as expected. To show that all simples in $\operatorname{\mathcal{H}}_S^\sharp$ are as in (3.1), take any simple T in $\operatorname{\mathcal{H}}_S^\sharp$ . Then $T\in{}^{\perp_{\operatorname{\mathcal{H}}}} S\subset\operatorname{\mathcal{H}}$ and we find a simple top X of T in $\operatorname{\mathcal{H}}$ and as above. We obtain exact sequences as in (3.6), which again leads to $T=\psi^\sharp_{S}(X)$ , concluding the argument.

Remark 3.3. The proposition above is a slight generalization of [Reference King and QiuKQ15, Proposition 5.2]. More precisely, when the simple S is rigid, new simples in the forward simple tilting formulae (3.1) and (3.2) are given by the simplified formulae

(3.7) \begin{eqnarray} {\psi}^{\sharp}_{S}(X)&=&\operatorname{Cone}(X\to S[1]\otimes\operatorname{Ext}^1(X, S)^* ) [-1],\end{eqnarray}

(3.8) \begin{eqnarray} {\psi}^{\flat}_{S}(X)&=&\operatorname{Cone} (S[-1] \otimes \operatorname{Ext}^1(S, X)\to X ).\end{eqnarray}

Inspecting the proof of Proposition 3.2, one finds that $\psi^\sharp_{S}(X)=\operatorname{Cone}(X\xrightarrow{f}\hat{S}[1])$ . More precisely, let $\operatorname{Ext}^1(X,S)=k$ correspond to the object $\hat{X}$ . Then we have $\operatorname{Ext}^1(\hat{X},S)=k$ , which corresponds to $\psi^\sharp_{S}(X)$ . Equivalently, we have the following tower of short exact sequences in $\operatorname{\mathcal{H}}$ (which arises from (3.4) with $M=X$ in $\operatorname{\mathcal{H}}^\sharp_{S}$ ).

4.1 Preliminaries on higher category theory

We formulate §§ 4, 5, and 7 in the language of Lurie’s stable $\infty$ -categories. In these sections, we will generally follow the notation of [Reference LurieLur09, Reference LurieLur17]. Given a stable $\infty$ -category, its homotopy 1-category inherits a triangulated structure. Because of their good formal properties, stable $\infty$ -categories thus provide us with a convenient choice of enhancement for triangulated categories. Even better formal properties are possessed by presentable, stable $\infty$ -categories. We will thus often work with the latter. Note that this is not much of a loss of generality: any stable $\infty$ -category $\mathcal{C}$ embeds fully faithfully into its $\operatorname{Ind}$ -completion $\operatorname{Ind}(\mathcal{C})$ , which is a presentable, stable $\infty$ -category. Furthermore, if $\mathcal{C}$ is idempotent complete, then the $\infty$ -category of compact objects $\operatorname{Ind}(\mathcal{C})^c$ in $\operatorname{Ind}(\mathcal{C})$ coincides with $\operatorname{\mathcal{C}}$ .

We will make use of the following $\infty$ -categories of $\infty$ -categories.

• – The $\infty$ -category of (not necessarily small) $\infty$ -categories, denoted by $\operatorname{Cat}_\infty$ .

• – The subcategory of $\operatorname{Cat}_\infty$ spanned by stable $\infty$ -categories and exact functors, denoted by $\operatorname{St}$ .

• – The subcategory of $\operatorname{Cat}_\infty$ spanned by presentable and stable $\infty$ -categories and left adjoint functors, denoted by $\mathcal{P}r^L_{\operatorname{St}}$ .

• – The subcategory of $\operatorname{Cat}_\infty$ spanned by presentable and stable $\infty$ -categories and right adjoint functors, denoted by $\mathcal{P}r^R_{\operatorname{St}}$ .

• – We let $\operatorname{LinCat}_k=\operatorname{LMod}_{\mathcal{D}(k)}(\mathcal{P}r^L_{\operatorname{St}})$ be the $\infty$ -category of k-linear $\infty$ -categories, consisting of left modules over the symmetric monoidal derived $\infty$ -category $\mathcal{D}(k)$ in $\mathcal{P}r^L_{\operatorname{St}}$ (see [Reference LurieLur18, Appendix D.1] for background).

For simplicity, we will only consider k-linear $\infty$ -categories in this paper, even though many concepts and constructions would also work R-linearly, with R any $\mathbb{E}_\infty$ -ring spectrum.

We note that passing to adjoints defines an equivalence of $\infty$ -categories

$$\begin{align}\operatorname{radj}(\text{-})\colon \mathcal{P}r^L_{\operatorname{St}}\simeq (\mathcal{P}r^R_{\operatorname{St}})^{\operatorname{op}} \mathpunct{} {:} \operatorname{ladj}(\text{-})\end{align}$$

(see [Reference LurieLur09, 5.5.3.4]). Recall also that a functor between presentable $\infty$ -categories admits a right adjoint (i.e. is a left adjoint) if and only if it preserves all (small) colimits (see [Reference LurieLur09, 5.5.2.9]).

Given a k-linear $\infty$ -category $\mathcal{C}$ , and two objects $X,Y\in \mathcal{C}$ , we denote by $\operatorname{RHom}_{\mathcal{C}}(X,Y)\in \mathcal{D}(k)$ the k-linear morphism object (see [Reference LurieLur17, 4.2.1.28]), characterized by the universal property

\[ \operatorname{Map}_{\mathcal{D}(k)}(C,\operatorname{RHom}_{\mathcal{C}}(X,Y))\simeq \operatorname{Map}_{\mathcal{C}}(C\otimes X,Y). \]

The remainder of this section concerns the computation of limits in $\infty$ -categories of $\infty$ -categories. The forgetful functors $\operatorname{LinCat}_k,\mathcal{P}r^L_{\operatorname{St}},\mathcal{P}r^R_{\operatorname{St}},\operatorname{St}\to \operatorname{Cat}_\infty$ preserve (small) limits. We remark that the forgetful functor $\operatorname{LinCat}_k\to \mathcal{P}r^L_{\operatorname{St}}$ also preserves small colimits (see [Reference LurieLur17, 4.2.3.5]). As we now recall, limits in $\operatorname{Cat}_\infty$ of strictly commuting diagrams of simplicial sets indexed by 1-categories admit a very explicit description in terms of sections of an explicit Grothendieck construction.

Let $F\colon Z\to \operatorname{Set}_{\Delta}$ be a functor from a 1-category Z to the 1-category of simplicial sets, that is, a strictly commuting diagram of simplicial sets. We suppose that F takes values in $\infty$ -categories, and thus gives rise to a diagram $\mathcal{F}\colon N(Z)\to \operatorname{Cat}_\infty$ (which one should think of as a homotopy commutative diagram), where N(Z) is the nerve of Z. In the following, we will not distinguish in notation between Z and N(Z).

Later on, we will be interested in the case where Z is the exit path category of a quiver, and thus all 2-simplices in Z will be degenerate. In this case, we thus have that any diagram $\mathcal{F}\colon Z\to \operatorname{Cat}_\infty$ commutes strictly, that is, arises from a functor F as above.

To F we associate an $\infty$ -category $\Gamma(F)$ , called the Grothendieck construction, together with a co-Cartesian fibration $p\colon \Gamma(F)\to Z$ , which is classified by the functor $\mathcal{F}$ . The definition of $\Gamma(F)$ is given in [Reference LurieLur09, 3.2.5.2], where $\Gamma(F)$ is called the relative nerve. Informally, an object in $\Gamma(F)$ consists of choices of $z\in Z$ and $x\in F(z)$ . A morphism in $\Gamma(F)$ from (z,x) to (z’,x’) consists of a morphism $\alpha\colon z\to z'$ in Z and a morphism $F(\alpha)(x)\to x'$ in F(z’).

A section of p consists of a functor $s\colon Z\to \Gamma(F)$ , whose composite with p is the identity on Z. These assemble into the $\infty$ -category of sections

\[ \operatorname{Fun}_Z(Z,\Gamma(F)):= \operatorname{Fun}(Z,\Gamma(F))\times_{\operatorname{Fun}(Z,Z)} \{\operatorname{id}_Z\}. \]

A section s is called co-Cartesian if s(e) is a p-co-Cartesian edge (in the dual sense of [Reference LurieLur09, 2.4.1.1]) for all edges e of Z. Concretely, an edge $s(e)\colon (z,x)\to (z',x')$ is p-co-Cartesian if and only if the morphism $F(\alpha)(x)\to x'$ in F(z’) is an equivalence. The full subcategory of $\operatorname{Fun}_Z(Z,\Gamma(F))$ spanned by co-Cartesian sections of p describes the limit of $\mathcal{F}$ (see [Reference LurieLur23, 7.4.1.9]). This description of the limit is both simple and very explicit, thus allowing for concrete computations. Computationally, we also benefit from the observation that the limit arises as the full subcategory of the bigger $\infty$ -category of all sections of p (the lax limit).

4.2 Spherical adjunctions and perverse schobers locally

In this section we discuss perverse schobers on ribbon graphs with a single vertex of valency n (n-spiders). These are shown to correspond to spherical adjunction, up to equivalence. They serve as the local model for perverse schobers on general ribbon graphs, which we consider in the subsequent section.

The notion of a spherical functor was introduced by Anno (see also [Reference Anno and LogvinenkoAL17]). Treatments in the setting of stable $\infty$ -categories can be found in [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24, Reference ChristChr22c].

As observed in [Reference Kapranov and SchechtmanKS18], spherical adjunctions may be seen as a categorification of perverse sheaves on the complex unit disc $\mathbb{D}\subset \mathbb{C}$ with a single singularity at 0. In this description, the $\infty$ -category $\mathcal{N}$ categorifies the generic stalk of the perverse sheaf, that is, any stalk away from 0, which is a chain complex concentrated in a single degree. The $\infty$ -category $\mathcal{V}$ categorifies the derived global sections of the perverse sheaf with support on the positive real axis $\mathbb{R}_{\geqslant 0}\cap \mathbb{D} \subset \mathbb{D}$ . These derived sections turn out to form a chain complex whose homology is concentrated in a single degree; this homology is called the vector space of vanishing cycles. Categorified perverse sheaves are also referred to as perverse schobers. We can think of $\mathbb{R}_{\geqslant 0} \cap \mathbb{D}\subset \mathbb{D}$ as describing a graph with a single, 1-valent vertex at 0 inside the disc. This graph is also called the 1-spider, as a special case of the j-spider (which we will use later).

We may thus summarize the above in the following definition.

Definition 4.3. A perverse schober parametrized by the 1-spider, or on the 1-spider for short, consists of a spherical adjunction between stable $\infty$ -categories

\begin{align}F\colon \mathcal{V}\longleftrightarrow \mathcal{N} \mathpunct{} {:} G.\end{align}

More generally, one can describe perverse sheaves on $\mathbb{D}$ in terms of their sections with support on the n-spider, where the n-valent vertex again lies at 0 and $n\geqslant 1$ (see [Reference Kapranov and SchechtmanKS16]). We propose to categorify this description as follows, leading to the notion of perverse schober on the n-spider.

Definition 4.4. Let $n\geqslant 2$ . A collection of adjunctions

\begin{align}(F_i\colon \mathcal{V}^n\longleftrightarrow \mathcal{N}_i {:} G_i)_{i\in \mathbb{Z}/n}\end{align}

between stable $\infty$ -categories is called a perverse schober parametrized by the n-spider, or a perverse schober on the n-spider for short, if:

1. (i°) $G_i$ is fully faithful, that is, $F_iG_i\simeq \operatorname{id}_{\mathcal{N}_i}$ via the counit;

2. (ii°) $F_{i}\circ G_{i+1}$ is an equivalence of $\infty$ -categories;

3. (iii°) $F_i\circ G_j\simeq 0$ if $j\neq i,i+1$ ;

4. (iv°) $G_i$ admits a right adjoint $\operatorname{radj}(G_i)$ ; and

5. (v°) $\operatorname{fib}(\operatorname{radj}(G_{i+1}))=\operatorname{fib}(F_{i})$ as full subcategories of $\mathcal{V}^n$ .

We will also consider a collection of functors $(F_i\colon\mathcal{V}^n\rightarrow \mathcal{N}_i)_{i\in \mathbb{Z}/n}$ as determining a perverse schober on the n-spider, or as a perverse schober on the n-spider for short, if there exist adjunctions $(F_i\dashv{\kern1pt} {\operatorname{radj}}(F_i))_{i\in \mathbb{Z}/n}$ which define a perverse schober on the n-spider.

In the subsequent sections of this paper, we will be interested in, and thus consider, k-linear perverse schobers, meaning that they take values in k-linear stable $\infty$ -categories (which we assumed to be presentable) and k-linear functors. However, the comparison of the above Definition 4.4 with the definition of a perverse schober on the n-spider given in [Reference ChristChr22a] does not rely on k-linearity or presentability, so we prefer to stay in the more general setting of arbitrary stable $\infty$ -categories in this subsection.

Remark 4.5. Consider a perverse schober on the n-spider $(F_i\colon \mathcal{V}^n\rightarrow \mathcal{N}_i)_{i\in \mathbb{Z}/n}$ . Since $G_i$ and $G_{i+1}$ are fully faithful, we find that their adjoints $F_i$ and $\operatorname{radj}(G_{i+1})$ are localizations in the sense that they factor through Verdier quotients as

\[F_i\colon \mathcal{V}^n\twoheadrightarrow\mathcal{V}^n\!/{\operatorname{fib}}(F_i)\simeq \mathcal{N}_i \]

and

\[ \operatorname{radj}(G_{i+1})\colon \mathcal{V}^n\twoheadrightarrow\mathcal{V}^n\!/{\operatorname{fib}}(\operatorname{radj}(G_{i+1}))\simeq \mathcal{N}_{i+1}. \]

By condition (v $^\circ$ ) in Definition 4.4, we have $\operatorname{fib}(F_i)=\operatorname{fib}(\operatorname{radj}(G_{i+1}))$ , and the resulting equivalence $\mathcal{N}_{i+1}\simeq \mathcal{N}_i$ identifies with $F_i G_{i+1}$ . We thus see that condition (ii $^\circ$ ) follows from condition (v $^\circ$ ), making it redundant. Furthermore, we find that $F_i\simeq F_i G_{i+1} \operatorname{radj}(G_{i+1})$ , meaning that $F_i$ and $\operatorname{radj}(G_{i+1})$ are equivalent up to postcomposition with the equivalence $F_i G_{i+1}\colon \mathcal{N}_{i+1}\simeq \mathcal{N}_{i}$ .

Note that this also shows that $F_i$ admits a left adjoint $\operatorname{ladj}(F_i)$ , which is then also equivalent to $G_{i+1}$ up to precomposition with an equivalence.

At first glance, Definition 4.4 may seem quite different from Definition 4.3. We proceed with justifying Definition 4.4, by showing that the datum of a perverse schober on the n-spider is equivalent to the datum of a perverse schober on the 1-spider. Before that, we illustrate the special case $n=2$ . In this case, the $\infty$ -category $\mathcal{V}^2$ admits a 4-periodic semiorthogonal decomposition $(\mathcal{V}^1,\mathcal{N})$ , which implies that the gluing functor $\mathcal{N}\to \mathcal{V}^1$ (see below) is spherical, giving rise to the corresponding perverse schober on the 1-spider. See also [Reference Halpern-Leistner and ShipmanHS16, Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24] for discussions of the relation between 4-periodic semiorthogonal decompositions and spherical functors. The two fully faithful functors $\mathcal{N}\simeq \mathcal{N}_i\xrightarrow{G_i} \mathcal{V}^2$ , $i=1,2$ , describe the inclusion of the component $\mathcal{N}$ of the semiorthogonal decomposition and the inclusion of a component of a mutated semiorthogonal decomposition.

Proposition 4.6. Let $n\geqslant 2$ . Given a perverse schober on the n-spider

$$\begin{align}(F_i\colon \mathcal{V}^n\longleftrightarrow \mathcal{N}_i \mathpunct{} {:}G_i)_{i\in \mathbb{Z}/n}\end{align}$$

and an integer $1\leqslant j\leqslant n$ , the collection of functors

(4.1) \begin{equation}(F_i|_{\operatorname{fib}(F_j)}\colon {\operatorname{fib}}(F_j)\longrightarrow \mathcal{N}_i)_{j\neq i\in \mathbb{Z}/n}\end{equation}

determines a perverse schober on the $(n-1)$ -spider.

We next describe the right adjoint of $F_{j+1}|_{\operatorname{fib}(F_j)}$ . The adjunction unit defines a natural transformation $\epsilon \colon G_{j+1}\to G_{j}F_{j}G_{j+1}$ , which becomes a natural equivalence after composing with $F_j$ . The image of the fibre $\operatorname{fib}(\epsilon)$ of $\epsilon$ is hence contained in $\operatorname{fib}(F_j)$ . We show that $\operatorname{fib}(\epsilon)$ is right adjoint to $F_{j+1}|_{\operatorname{fib}(F_j)}$ . We consider the adjunction unit $\operatorname{u}\colon {\operatorname{id}}_{\mathcal{V}^n}\to G_{j+1} F_{j+1}$ . Any natural transformation $\operatorname{id}_{\operatorname{fib}(F_j)}\to G_jF_jG_{j+1}F_{j+1}$ is adjoint to a natural transformation $0\simeq F_j|_{{\operatorname{fib}(F_j)}}\to F_jG_{j+1}F_{j+1}$ and thus vanishes. Thus, the restriction of $\operatorname{u}$ to ${\operatorname{fib}(F_j)}$ induces a natural transformation

\[\operatorname{u}'\colon {\operatorname{id}}_{\operatorname{fib}(F_j)}\to \operatorname{fib}(\epsilon)\circ F_{j+1}|_{\operatorname{fib}(F_j)}\simeq \operatorname{fib}(G_{j+1}F_{j+1}\to G_{j}F_{j}G_{j+1}F_{j+1}). \]

We define a second natural transformation as the natural equivalence

\[\operatorname{cu}'\colon F_{j+1}|_{{\operatorname{fib}(F_j)}} \circ\operatorname{fib}(\epsilon)\simeq F_{j+1} G_{j+1}\simeq\operatorname{id}_{\mathcal{N}_{j+1}}\!. \]

The natural transformations $\operatorname{u}'$ and $\operatorname{cu}'$ satisfy the triangle identities.

The natural transformation $F_{j+1}|_{\operatorname{fib}(F_j)}\operatorname{u}'$ identifies with the restriction of $F_{j+1}\operatorname{u}$ since $F_{j+1} G_j\simeq 0$ . The composite $(\operatorname{cu}'F_{j+1}|_{\operatorname{fib}(\epsilon)})\circ (F_{j+1}|_{\operatorname{fib}(\epsilon)}\operatorname{u}')$ thus identifies with the restriction of the natural transformation $\operatorname{id}_{F_{j+1}}\simeq (\operatorname{cu}F_{j+1})\circ (F_{j+1}\operatorname{u})\colon F_{j+1}\simeq F_{j+1}$ to ${\operatorname{fib}(F_j)}$ .

As above, any natural transformation

\[ \operatorname{fib}(\epsilon)\simeq \operatorname{id}_{\operatorname{fib}(F_j)}\circ \operatorname{fib}(\epsilon)\to G_jF_jG_{j+1} \]

is adjoint to a zero transformation and thus vanishes. Hence, to show that $(\operatorname{fib}(\epsilon) \operatorname{cu}')\circ (\operatorname{u}' \operatorname{fib}(\epsilon))$ is equivalent to the identity, it suffices to show that $(\operatorname{fib}(\epsilon) \operatorname{cu}')\circ (\operatorname{u}' \operatorname{fib}(\epsilon))$ composes with the fibre natural transformation $\alpha\colon {\operatorname{fib}}(\epsilon)\to G_{j+1}$ of $\epsilon$ to the fibre natural transformation $\alpha$ . This follows from the commutativity of the following diagram.

The commutativity of the top and bottom triangles follows from the definition of $\operatorname{cu}'$ , $\operatorname{u}'$ , and the triangle identity for $F_{j+1}\dashv{\kern1pt} G_{j+1}$ . The commutativity of the left and right squares is the interchange law for natural transformations and the middle square trivially commutes.

We have thus proved the two triangle identities, showing that $F_{j+1}|_{\operatorname{fib}(\epsilon)}$ is left adjoint to $\operatorname{fib}(\epsilon)$ (see [Reference LurieLur23, Tags 02EK,02EL]).

With the above, we verify in what follows that the collection of functors (4.1) and their right adjoints satisfy the conditions of Definition 4.4. For $i\in (\mathbb{Z}/n)\backslash \{j\}$ , let $G_i'$ denote the right adjoint of $F_i|_{\operatorname{fib}(F_j)}$ . For $i\not= j+1$ , $G_i'$ is a restriction of $G_i$ and thus fully faithful. The functor $G_{j+1}'=\operatorname{fib}(\epsilon)$ is fully faithful since the adjunction counit $\operatorname{cu}'$ is an equivalence. This shows condition (i $^\circ$ ).

For condition (ii $^\circ$ ), the only new condition to check is that $F_{j-1}|_{\operatorname{fib}(F_j)}\circ G_{j+1}'$ is an equivalence. This follows from

\[ F_{j-1}|_{\operatorname{fib}(F_j)}\circ \operatorname{fib}(\epsilon)\simeq \operatorname{fib}(\underbrace{F_{j-1}\circ G_{j+1}}_{\simeq 0}\to F_{j-1}G_jF_jG_{j+1})\simeq F_{j-1}G_jF_jG_{j+1}[-1], \]

where $F_{j-1}G_j$ and $F_jG_{j+1}$ are each equivalences by assumption.

For condition (iii $^\circ$ ), the new condition to check is that $F_i|_{\operatorname{fib}(F_j)}\circ \operatorname{fib}(\epsilon)\simeq 0$ for $i\not= j+1,j-1$ , which, by inspecting the fibre sequence defining $\operatorname{fib}(\epsilon)$ , reduces to $F_i\circ G_{j+1},F_i\circ G_{j}\simeq 0$ .

For condition (iv $^\circ$ ), we note that the right adjoint of $G_{i}'$ is given by $\operatorname{radj}(G_i)|_{\operatorname{fib}(F_j)}$ for $i\not= j+1$ . Since the passage to right adjoints defines an exact functor, $\operatorname{fib}(\epsilon)$ admits a right adjoint, given by $\operatorname{cof}(\operatorname{radj}(G_{j+1})G_j\operatorname{radj}(G_j)\to \operatorname{radj}(G_{j+1}))$ .

For condition (v $^\circ$ ), the only new condition to check is that the fibre of $\operatorname{radj}(G_{j+1}')=\operatorname{radj}(\operatorname{fib}(\epsilon))$ agrees with $\operatorname{fib}(F_{j-1}|_{\operatorname{fib}(F_j)})$ . We observe that

\[ \operatorname{radj}(G_{j+1}')\simeq \operatorname{cof}(\operatorname{radj}(G_{j+1})G_j\operatorname{radj}(G_j)\to \operatorname{radj}(G_{j+1})) \]

is equivalent to $\operatorname{radj}(G_{j+1})G_j\operatorname{radj}(G_j')[1]$ , since $\operatorname{radj}(G_{j+1})$ vanishes on $\operatorname{fib}(F_j)=\operatorname{fib}(\operatorname{radj}(G_{j+1}))$ (where the latter equality arises from condition (v $^\circ$ ) for the perverse schober on the n-spider). The assumption that $F_jG_{j+1}$ is an equivalence implies, after passing to right adjoints, that $\operatorname{radj}(G_{j+1})G_j$ is also an equivalence. It follows that

\[\operatorname{fib}(\operatorname{radj}(G_{j+1}'))\simeq \operatorname{fib}(\operatorname{radj}(\operatorname{G_j})|_{\operatorname{fib}(F_j)})\simeq \operatorname{fib}(F_{j-1}|_{\operatorname{fib}(F_j)}), \]

again by condition (v $^\circ$ ) of the perverse schober on the n-spider.

We proceed with the case $n=2$ . We prove the assertion via a different argument than above, but note for completeness that the right adjoint of $F_2|_{\operatorname{fib}(F_1)}$ is again given by $\operatorname{fib}(\epsilon)=\operatorname{fib}(G_2\to G_1F_1G_2)$ . Using the adjunctions $F_1\dashv{\kern1pt} G_1$ and $F_2\dashv{\kern1pt} G_2$ and [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24, Proposition 2.3.2], we find that there are semiorthogonal decompositions $(\operatorname{Im}(G_i),\operatorname{fib}(F_i))$ and $(\operatorname{fib}(\operatorname{radj}(G_i)),\operatorname{Im}(G_i))$ for $i=1,2$ , where $\operatorname{Im}(G_i)\subset \mathcal{V}^n$ denotes the essential image of $G_i$ . The condition $\operatorname{fib}(\operatorname{radj}(G_{i+1}))=\operatorname{fib}(F_i)$ is equivalent to the 4-periodicity of these semiorthogonal decompositions, in the sense of [Reference Halpern-Leistner and ShipmanHS16, Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24]. It follows from [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24, Proposition 2.5.12] that the gluing functor of $(\operatorname{fib}(\operatorname{radj}(G_i)),\operatorname{Im}(G_i))$ , given by the restriction of $F_i$ to $\operatorname{fib}(\operatorname{radj}(G_i))=\operatorname{fib}(F_{i-1})$ , is a spherical functor.

Next, we describe an explicit model for perverse schobers on the n-spider, taking as input a spherical functor. This model is inspired by Dyckerhoff’s categorified Dold–Kan correspondence [Reference DyckerhoffDyc21] (see also the discussion in [Reference ChristChr22b]). Let $F\colon \mathcal{V}\to \mathcal{N}$ be a spherical functor and define $\mathcal{V}^n_F$ as the stable $\infty$ -category of diagrams

\[ a\rightarrow b_1\rightarrow \cdots \rightarrow b_{n-1} \]

where $a\in \mathcal{V}$ , $b_i\in \mathcal{N}$ , and the morphism $a\rightarrow b_1$ lies in the Grothendieck construction of F, that is, encodes a morphism $F(a)\rightarrow b_1$ . For a more precise definition of $\mathcal{V}^n_F$ , in terms of sections of a Grothendieck construction, see [Reference ChristChr22b, Notation 3.2, Lemma 2.29]. We define $\varrho_1=\pi_{n-1}\colon \mathcal{V}^n_F\rightarrow \mathcal{N}$ to be the projection functor to $b_{n-1}$ and, for $2\leqslant i\leqslant n$ , $\varrho_i$ to be the $(2i-2)$ -fold left adjoint of $\pi_{n-1}$ . On objects, the functor $\varrho_i$ can be explicitly described via the following formula (see [Reference ChristChr22b, Lemma 3.3]). Let $X=(a\to b_1\to \dots \to b_{n-1})\in \mathcal{V}^n_F$ . Then

(4.2) \begin{equation} \varrho_i(X)\simeq \begin{cases} b_{n-1}, & i=1, \\ \operatorname{fib}(b_{n-i}\to b_{n-i+1})[i-1], & 1\lt i\lt n,\\ \operatorname{fib}(F(a)\to b_1)[n-1], & i=n. \end{cases}\end{equation}

The functor $\varrho_i$ admits a right adjoint $\varsigma_i\colon \mathcal{N}\to \mathcal{V}^n_F$ . This functor can be described just as explicitly on objects (see again [Reference ChristChr22b, Lemma 3.3]). Let $b\in \mathcal{N}$ . Then

(4.3)

where the above morphism $G(b)\to b$ is Cartesian in the Grothendieck construction, meaning it encodes the counit morphism $FG(b)\to b$ .

The collection of adjunctions $(\varrho_i\dashv{\kern1pt}\varsigma_i)_{i\in \mathbb{Z}/n}$ defines a perverse schober on the n-spider, as follows readily from inspecting the above descriptions and using [Reference ChristChr22b, Lemma 3.8].

Proposition 4.7 (The model for perverse schobers on the n-spider). Let $F\colon \mathcal{V}\to \mathcal{N}$ be a spherical functor. The collection of adjunctions

\begin{align}(\varrho_i\colon \mathcal{V}^n_F\longleftrightarrow\mathcal{N} {:}\varsigma_i)_{i\in \mathbb{Z}/n}\end{align}

defines a perverse schober on the n-spider, denoted by $\mathcal{F}_n(F)$ .

Consider a stable $\infty$ -category $\mathcal{C}$ with a semiorthogonal decomposition $(\mathcal{A},\mathcal{B})$ . The inclusion $\iota_{\mathcal{A}}\colon \mathcal{A}\subset\operatorname{\mathcal{C}}$ always admits a left adjoint and the inclusion $\iota_{\mathcal{B}}\colon \mathcal{B}\subset \operatorname{\mathcal{D}}$ always admits a right adjoint [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24, Proposition 2.3.2]. Furthermore, if $\iota_{\mathcal{A}}$ admits a right adjoint $\iota_{\mathcal{A}}^R$ , then $(\mathcal{A},\mathcal{B})$ is called a Cartesian semiorthogonal decomposition and the corresponding gluing functor is defined as $\iota_A^R\circ \iota_{\mathcal{B}}\colon \mathcal{B}\to \mathcal{A}$ . Similarly, if $\iota_{\mathcal{B}}$ admits a left adjoint $\iota_{\mathcal{B}}^L$ , then $(\mathcal{A},\mathcal{B})$ is called a co-Cartesian semiorthogonal decomposition, and the gluing functor is $\iota_{\mathcal{B}}^L\circ \iota_{\mathcal{A}}\colon \mathcal{A}\to \mathcal{B}$ . In either of these two cases, $\operatorname{\mathcal{C}}$ can be reconstructed given $\mathcal{A}$ and $\mathcal{B}$ as well as the gluing functor (see [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24, Remark 2.3.7]).

Given a perverse schober on the n-spider $\mathcal{F}$ , we obtain the spherical adjunction underlying $\mathcal{F}$ , by repeatedly applying Proposition 4.6 to any $n-1$ of the n edges of the n-spider. Up to equivalence, the arising spherical adjunction is independent of the choice of edges.

Suppose that $n\gt1$ . We denote the perverse schober $\mathcal{F}$ by $(F_i\colon \mathcal{V}^n\longleftrightarrow \mathcal{N}_i \mathpunct{} {:} G_i)_{i\in \mathbb{Z}/n}$ . Fix $j\in \mathbb{Z}/n$ and denote the perverse schober on the $(n-1)$ -spider arising from $\mathcal{F}$ via Proposition 4.6 by $(F_i'\colon \mathcal{V}^{n-1}\longleftrightarrow \mathcal{N}_i \mathpunct{} {:} G_i')_{i\in \mathbb{Z}/n, i\not= j}$ . For the induction step, assume that $\mathcal{F}'\simeq \mathcal{F}_{n-1}(F)$ . Then $\mathcal{V}^{n-1}=\operatorname{fib}(F_j)\simeq \mathcal{V}^{n-1}_F$ . The stable $\infty$ -category $\mathcal{V}^n$ has a semiorthogonal decomposition into $\mathcal{V}^{n-1}$ and $\mathcal{N}_j\simeq \mathcal{N}$ with fully faithful gluing functor (see Lemma 4.9 below). Furthermore, $\mathcal{V}^n_F$ has a semiorthogonal decomposition into $\mathcal{V}^{n-1}_F$ and $\mathcal{N}$ , and the corresponding gluing functor identifies with the gluing functor of $\mathcal{V}^n$ . We thus find $\mathcal{V}^n\simeq \mathcal{V}^n_F$ . Further still, under this identification with $\mathcal{V}^n_F$ , the functor $F_j$ corresponds to the left adjoint of the inclusion of the component $\mathcal{N}\simeq \mathcal{N}_j$ , meaning to $\varrho_j$ . By Remark 4.5, the functors $F_i$ and $\varrho_i$ , with $i\in (\mathbb{Z}/n)\backslash \{j\}$ , can be recovered up to postcomposition with equivalences by passing to repeated adjoints of $F_j$ and $\varrho_j$ , respectively. Thus also $F_i$ is equivalent to $\varrho_i$ (up to postcomposition with equivalences), showing the desired equivalence of perverse schobers.

In the case $n=2$ , the semiorthogonal decomposition is 4-periodic, as shown in the proof of Proposition 4.6, so that the gluing functor is spherical by [Reference Dyckerhoff, Kapranov, Schechtman and SoibelmanDKS⁺24, Proposition 2.5.5].

We next consider the case $n\geqslant 3$ . Denote by $\iota$ the fully faithful functor $\operatorname{fib}(F_{i})\to \mathcal{V}^n$ . The gluing functor $\mathcal{N}_{i+1}\xrightarrow{G_{i+1}}\mathcal{V}^n\xrightarrow{\operatorname{radj}(\iota)}\operatorname{fib}(F_{i})$ is the right adjoint of $F_{i+1}|_{\operatorname{fib}(F_{i})}$ , and thus fully faithful by Proposition 4.6.

4.3 Parametrized perverse schobers

In this section, we recall the definition of a perverse schober parametrized by a ribbon graph in terms of certain functors from the exit path category of the graph, as well as the $\infty$ -categories of global sections and lax sections.

Let v be a vertex of valency n of a ribbon graph ${\mathbf{G}}$ . Let $\operatorname{Exit}({\mathbf{G}})_{v/}$ be the undercategory, which has $n+1$ objects, which can be identified with v and its n incident half-edges and non-identity morphisms going from v to these half-edges. There is a functor $\operatorname{Exit}({\mathbf{G}})_{v/}\to \operatorname{Exit}({\mathbf{G}})$ , which is fully faithful if $\Gamma$ has no loops incident to v.

Given a connected ribbon graph ${\mathbf{G}}$ , a ${\mathbf{G}}$ -parametrized perverse schober $\mathcal{F}$ assigns to each edge of ${\mathbf{G}}$ an equivalent stable $\infty$ -category, referred to as the generic stalk of $\mathcal{F}$ .

The local rotational ‘symmetry’ of a perverse schober at a vertex is captured by composition with the equivalence $T_{\mathcal{F}(v)}$ defined in the following proposition. This is in general not a genuine symmetry, as a full rotation does not return the perverse schober to itself; the change depends on the ‘monodromy’ around the singularity given by the suspension of the cotwist functor of the corresponding spherical adjunction. We will refer to this rotational action as the paracyclic symmetry (see also [Reference ChristChr22b] for background).

Proposition 4.15 [Reference ChristChr22b, Lemma 3.8]. Let $\mathcal{F}$ be a ${\mathbf{G}}$ -parametrized perverse schober and $v\in {\mathbf{G}}_0$ a vertex with incident half-edges $a_1,\dots,a_n$ lying in edges $e_1,\dots,e_n\in {\mathbf{G}}_1$ . The functor

\[ F_v':= \prod_{i=1}^n\mathcal{F}(v\xrightarrow{a_i}e_i)\colon \mathcal{F}(v)\longrightarrow \prod_{i=1}^n\mathcal{F}(e_i) \]

is spherical. Let $G_v'$ be its right adjoint. We denote the twist functor of the adjunction $F_v'\dashv{\kern1pt} G_v'$ by $T_{\mathcal{F}(v)}$ .

4.4 Equivalences from contractions of ribbon graphs

To complement the discussion in the previous section, we recall here the fact that one can transport perverse schobers along certain contractions of ribbon graphs in a way that preserves the $\infty$ -category of global sections, up to equivalence.

Let ${\mathbf{G}}$ be a ribbon graph and $e\in {\mathbf{G}}_1^\circ$ an internal edge which is not a loop. By contracting the edge e, we obtain a new ribbon graph ${\mathbf{G}}'$ , which we call the contraction of ${\mathbf{G}}$ at the edge e. For a more formal description of this procedure, see [Reference ChristChr22b, Definition 4.24]. Given two ribbon graphs ${\mathbf{G}}$ and ${\mathbf{G}}'$ , we write $c\colon {\mathbf{G}}\rightarrow {\mathbf{G}}'$ if ${\mathbf{G}}'$ can be obtained from ${\mathbf{G}}$ by contracting finitely many edges, and call c a contraction of ribbon graphs.

For part (2), it suffices to treat the case that c contracts a single edge e of ${\mathbf{G}}$ incident to vertices $v_1,v_2$ , whose image in ${\mathbf{G}}'$ is denoted v. Swapping the labels of $v_1$ and $v_2$ if necessary, we may assume that $S=\{v_2\}$ . Let $m_i$ be the valency of $v_i$ , $i=1,2$ , and $n=m_1+m_2-2$ the valency of v. Replacing $\mathcal{F}'$ by an equivalent ${\mathbf{G}}'$ -parametrized perverse schober, we may assume that near v, $\mathcal{F}'$ is given by the following diagram, with $F_v'\colon \mathcal{V}\to \mathcal{N}$ the spherical functor underlying $\mathcal{F}'$ at v (see also Proposition 4.8).

We define $c^*(\mathcal{F}')$ to be identical to $\mathcal{F}'$ away from $v_1,v_2$ and near $v_1,v_2$ as in the following diagram:

where we denote by $0_{\mathcal{N}}$ the spherical functor $0\colon 0\rightarrow \mathcal{V}_v$ . By [Reference ChristChr22b, Lemma 4.26], we then get $c_*c^*(\mathcal{F}')\simeq \mathcal{F}'$ . The equivalence ${\Gamma}({\mathbf{G}}',\mathcal{F}')\simeq {\Gamma}({\mathbf{G}},c^*(\mathcal{F}'))$ in part (2) thus follows from part (1).

5.1 Equivalences from flips of S-graphs

In this section we explain how to realize flips of S-graphs in terms of zig-zags of edge contractions, thereby allowing perverse schobers to be transported across such flips in a way that preserves their $\infty$ -categories of global sections, up to equivalence.

We refer to the half-edges and edges of $\widehat{\mathbb{S}}$ which are not part of $\mathbb{S}$ as virtual. These edges are not embedded in $\mathbf{S}_{\mathbf{w}}$ . The primary reason for their introduction is rather technical: the diagrams we wish to study define $\widehat{\mathbb{S}}$ -parametrized perverse schobers, but not $\mathbb{S}$ -parametrized perverse schobers. The other half-edges and edges of $\widehat{\mathbb{S}}$ are called non-virtual and can be identified with the half-edges of edges of $\mathbb{S}$ . We also identify the vertices of $\widehat{\mathbb{S}}$ and $\mathbb{S}$ , and use this when referring to vertices of $\widehat{\mathbb{S}}$ as internal or boundary vertices.

In the case that $\eta$ is incident to a single singular point of weight $-1$ (i.e. degree 1), we instead use the zig-zag of contractions depicted in Figure 7.

Figure 6.

A span of contractions moving a half-edge, describing part of the forward flip at a normal arc.

Figure 7.

The zig-zag of contractions describes a forward flip at a monogon arc.

5.2 Arc system kits

A well-known construction in the symplectic geometry of Lefschetz fibrations, originally due to Donaldson, and explained in detail in [Reference SeidelSei08], yields Lagrangian matching spheres (also known as matching cycles) using vanishing cycles and matching paths as input. A vanishing cycle refers to a Lagrangian sphere in the symplectic fibre of the Lefschetz fibration, which collapses to a point when parallel-transported along an arc in the base ending in a singular value of the fibration. If we can find two such singular values where the parallel transport of the Lagrangian sphere collapses, the union of the Lagrangian sphere and all its parallel transports in all fibres over the arc connecting the two singular values forms a Lagrangian sphere (of one dimension higher) in the total space of the fibration, referred to as the matching sphere (see Figure 8). This construction has an analogue in the setting of perverse schobers: consider an edge e of a ribbon graph ${\mathbf{G}}$ connecting the two vertices $v_1,v_2$ . Let $\mathcal{F}$ be a ${\mathbf{G}}$ -parametrized perverse schober. The vanishing cycle corresponds to a choice of object $L_e\in \mathcal{F}(e)$ . In a typical situation (see, for instance, the perverse schobers considered in [Reference ChristChr22b, Reference ChristChr21]), $L_e$ would be a spherical object, meaning that the functor $\text{-}\otimes L_e\colon {\operatorname{\mathcal{D}}}(k)\to \mathcal{F}(e)$ is a spherical functor. The categorical meaning of $L_e$ being a vanishing cycle that collapses at either end of the arc e is that this spherical functor $\text{-}\otimes L_e$ describes the perverse schober $\mathcal{F}$ at $v_1$ and $v_2$ . There are then distinguished objects $L_{v_1,e}\in \mathcal{F}(v_1),L_{v_2,e}\in \mathcal{F}(v_2)$ , satisfying that $\mathcal{F}(v_1\to e)(L_{v_1,e})\simeq L_e$ and $\mathcal{F}(v_2\to e)(L_{v_2,e})\simeq L_e$ . Note that if $v_i$ is 1-valent, then simply $L_{v_i,e}=k\in \operatorname{\mathcal{D}}(k)=\mathcal{F}(v_i)$ . We can consider $L_e,L_{v_1,e},L_{v_2,e}$ as compatible lax sections of the perverse schober, which thus glue together to a global section of $\mathcal{F}$ , which is the analogue of the matching sphere.

Figure 8.

Arc in the base surface and corresponding matching sphere in the total space of the fibration.

We next consider the generalized, categorical analogue of the (quite favourable) situation where the fibres of the Lefschetz fibration are each equipped with a choice of a vanishing cycle, these being related to each other via parallel transport, and such that the vanishing cycles collapse over every singular value of the fibration. In this situation, we can associate matching spheres with arbitrary graded arcs in the base surface of the fibration. The corresponding categorical data for a perverse schober (see Definition 5.4), is referred to as an arc system kit. The additional condition (v) in Definition 5.4 essentially ensures that there is only a single object (up to shift) taking the role of the vanishing cycle near each vertex, but furthermore states that the arc system kit has a local rotational symmetry similar to the rotational symmetry of the polygons.

When considering arc system kits for an $\widehat{\mathbb{S}}$ -parametrized perverse schober $\mathcal{F}$ , we will always assume that the set of singularities of $\mathcal{F}$ consists exactly of the interior vertices of $\widehat{\mathbb{S}}$ . We fix such an $\widehat{\mathbb{S}}$ -parametrized perverse schober $\mathcal{F}$ .

Note that in particular, if v is q-valent of degree m, one has $T_{\mathcal{F}(v)}^m(L_{v,a})\simeq L_{v,a}[q-m]$ .

The datum in an arc system kit satisfies a ‘local-to-global principle’, meaning that it amounts to local arc system kits, one for each vertex of $\widehat{\mathbb{S}}$ and its incident edges, plus the requirement that the data at the edges agrees. The construction below describes the default local arc system kits that we will use.

Consider the disc with a single interior singular point v of degree $m\geqslant q$ and q singular points on the boundary. There is a mixed angulation with dual graph $\mathbb{S}$ having a single q-valent interior vertex v and all interior edges emanating from v. We choose any total order on these q interior edges which is compatible with their given cyclic order, labelling them $e_1,\dots,e_q$ . Given a spherical functor $F\colon \mathcal{V}\to \mathcal{N}$ , we let $\mathcal{F}_v(F)$ be the $\operatorname{\mathbf{Sp}}_{q}$ -parametrized perverse schober obtained from $\mathcal{F}_q(F)$ , as in Proposition 4.7, by composing $\mathcal{F}_q(F)(v\to e_i)$ with $[d(a_1,a_i)-(i-1)]$ for all $1\leqslant i\leqslant q$ . Note that if $q=m$ , then $\mathcal{F}_v(F)=\mathcal{F}_q(F)$ . Let $L_v\in \mathcal{V}$ be an object satisfying that $T_{\mathcal{V}}(L_v)\simeq L_v[1-m]$ , with $T_{\mathcal{V}}$ the twist functor of $F\dashv G$ . For instance, if $\mathcal{V}=\operatorname{\mathcal{D}}(k)$ , $L_v=k\in \operatorname{\mathcal{D}}(k)$ , and $F(k)\in \mathcal{N}$ is an m-spherical object, then $T_{\mathcal{V}}(L_v)\simeq L_v[1-m]$ holds. The data for an arc system kit for $\mathcal{F}_v(F)$ at the vertex v is obtained as follows.

We set $L_e=F(L_v)\in \mathcal{N}=\mathcal{F}_v(F)(e)$ for all edges $e\in \widehat{\mathbb{S}}_1$ . Let $a_i$ be the ith half-edge incident to v. We set $L_{v,a_i}\in \mathcal{V}_{F}^q=\mathcal{F}_v(F)(v)$ to be the diagram

(5.2) \begin{equation}\bigg({\kern-1pt}L_v\rightarrow F(L_v)\xrightarrow{\operatorname{id}}F(L_v)\xrightarrow{\operatorname{id}}\cdots\xrightarrow{\operatorname{id}}\underbrace{F(L_v)}_{(q-i+1)\text{th}}\rightarrow 0\rightarrow \cdots \rightarrow 0\!\bigg)[-d(a_1,a_i)]\end{equation}

where the first morphism $L_v\to F(L_v)$ is co-Cartesian in the Grothendieck construction of F. Then we have

\[\mathcal{F}_q(F)(v\to e_1)(L_{v,a_i})\simeq\varrho_1(L_{v,a_i})=\pi_{q-1}(L_{v,a_i})\simeq \begin{cases} F(L_v),& a=1, \\ 0, & a\gt1 . \end{cases} \]

Let $2\leqslant j\leqslant n-1$ . Using the equivalence $\varrho_j\simeq \operatorname{fib}_{n-j,n-j+1}[j-1]$ from equation (4.2), we find

\[ \mathcal{F}_v(F)(v\to e_j)(L_{v,a_i})\simeq \varrho_j(L_{v,a_i})[d(a_1,a_j)-(j-1)]\simeq \operatorname{fib}_{n-j,n-j+1}(L_{v,a_i})[d(a_1,a_j)], \]

where the latter object indicates by definition the ( $[d(a_1,a_j)]$ -shifted) fibre of the morphism between the $(n-j)$ th and $(n-j+1)$ th entries in diagram (5.2). We thus find

\[\mathcal{F}_v(F)(v\to e_j)(L_{v,a_i})\simeq \begin{cases}\operatorname{fib}(F(L_v)\xrightarrow{\operatorname{id}}F(L_v))[d(a_1,a_j)-d(a_1,a_i)]\simeq 0, & j\gt i, \\\operatorname{fib}(F(L_v)\to 0)\simeq F(L_v), & j=i, \\\operatorname{fib}(0\to 0)\simeq 0, &j\lt i.\end{cases} \]

The last functor $\varrho_q$ corresponds to taking the relative fibre along the morphism in the Grothendieck construction, as in equation (4.2), meaning that

\begin{align*}\mathcal{F}_v(F)(v\to e_q)(L_{v,a_i})& \simeq \varrho_q(L_{v,a_i})[d(a_1,a_q)-(q-1)](L_{v,a_i})\\& \simeq \begin{cases} \operatorname{fib}(F(L_v)\to F(L_v))[d(a_1,a_q)-d(a_1,a_i)]\simeq 0, & i\lt q, \\\operatorname{fib}(F(L_v)\to 0)\simeq F(L_v), & i=q.\end{cases}\end{align*}

This shows the equivalences as in (iii) in the definition of an arc system kit. We next show the equivalence (5.1) from (v). Let $\mathcal{F}_v(F)(v\to e_i)^R$ be the right adjoint of $\mathcal{F}_v(F)(v\to e_i)$ . We then have $\prod_{1\leqslant j\leqslant q} \mathcal{F}_v(F)(v\to e_j)^R \mathcal{F}_v(F)(v\to e_j) (L_{v,a_i})\simeq \mathcal{F}_v(F)(v\to e_i)^R(F(L_v))\in \mathcal{F}_v(F)(v)$ . This object can be depicted for $i\gt1$ as

\[\bigg({\kern-1pt}0\to \dots \to 0\to\underbrace{F(L_v)}_{(q-i+2)\text{th}} \to 0\to\dots \to 0\!\bigg)[-d(a_1,a_i)+1] \]

and for $i=1$ as

\[( GF(L_v)\to F(L_v)\xrightarrow{\operatorname{id}}\cdots \xrightarrow{\operatorname{id}}F(L_v)), \]

as follows from equation (4.3) and the definition of $\mathcal{F}_v(F)$ . For $i=1$ , we can depict the resulting (vertical) cofibre sequence computing $T_{\mathcal{F}_v(F)(v)}(L_{v,a_i})\in \mathcal{F}_v(F)(v)$ as follows (note that the cofibre is computed ‘entrywise in the diagram’).

Since $T_{\mathcal{V}}(L_v)\simeq L_v[1-m]$ and $m=d(a_1,a_q)+d(a_q,a_1)$ , we find $T_{\mathcal{F}_v(F)(v)}(L_{v,a_1})\simeq L_{v,a_{q}}[1-d(a_{q},a_1)]$ . For $i\gt1$ , we similarly see that $T_{\mathcal{F}_v(F)(v)}(L_{v,a_i})\simeq L_{v,a_{i-1}}[1-d(a_{i-1},a_i)]$ .

Case II: Boundary vertices.

Consider the disc with q singular points on the boundary and with a mixed angulation whose dual S-graph $\mathbb{S}$ consists of q edges emanating from a single boundary vertex v. At v, the extended S-graph $\widehat{\mathbb{S}}$ is described by the $(q+1)$ -spider. The half-edges of the $(q+1)$ -spider are equipped with the total order where the virtual half-edge is in the final position. Let $\mathcal{F}$ be an $\widehat{\mathbb{S}}$ -parametrized perverse schober which restricts on the $(q+1)$ -spider to the perverse schober $\mathcal{F}_v(0)$ obtained from $\mathcal{F}_q(0)$ , with $0\colon 0\rightarrow \mathcal{N}$ the zero functor, by composing $\mathcal{F}_q(0)(v\to e_i)$ with $[d(a_1,a_i)-(2i-1)]$ for $1\leqslant i\leqslant q$ . Note that $\mathcal{F}_v(0)(v)= \mathcal{V}_0^{q+1}\simeq \operatorname{Fun}(\Delta^{q-1},\mathcal{N})$ . Choose any object $L\in \mathcal{N}$ . We set $L_e=L\in \mathcal{F}(e)=\mathcal{N}$ for all non-virtual edges e of $\widehat{\mathbb{S}}$ . We further set $L_{v,a_i}\in \mathcal{V}_0^{q+1}$ to be the object

(5.3) \begin{equation}\bigg(\!L\xrightarrow{\operatorname{id}}L\xrightarrow{\operatorname{id}}\cdots\xrightarrow{\operatorname{id}}\underbrace{L}_{(q-i+1)\text{th}}\rightarrow 0\rightarrow \cdots \rightarrow 0\!\bigg)[i-d(a_1,a_i)].\end{equation}

A computation similar to the above again shows that these objects describe the data of an arc system kit for $\mathcal{F}$ at v.

In the following, we assume that $\mathcal{F}$ is equipped with an arc system kit.

Proof. We begin by showing part (1). Since $\mathbf{S}_{\mathbf{w}}$ is assumed to be connected, it suffices to consider the case where e,h are incident to the same vertex v with $a\in e$ and $b\in h$ with b the successor half-edge of a at v. Let $T_{\mathcal{F}(v)}$ be the twist functor of the adjunction $F_v'\dashv{\kern1pt} G_v'$ from Proposition 4.15. There exists an equivalence of functors

\[ \mathcal{F}(v\xrightarrow{a}e)\circ T_{\mathcal{F}(v)}\simeq T'\circ \mathcal{F}(v\xrightarrow{b}h) \]

with $T'\colon \mathcal{F}(h)\simeq \mathcal{F}(e)$ some equivalence. This follows from the fact that, at v, $\mathcal{F}$ is locally equivalent to a perverse schober on the n-spider $\mathcal{F}_n(F)$ of the form described in Proposition 4.7 (see Proposition 4.8). For $\mathcal{F}_n$ , one has $\mathcal{F}_n(F)(v\xrightarrow{a}e)\circ T_{\mathcal{F}_n(F)(v)}\simeq T''\circ \mathcal{F}_n(F)(v\xrightarrow{b}h)$ for some equivalence T” by [Reference ChristChr22b, Proposition 3.11]. We thus find

\begin{align*}T'(L_{h})& \simeq T'\circ \mathcal{F}(v\xrightarrow{b}h)(L_{v,b}) \simeq \mathcal{F}(v\xrightarrow{a}e)\circ T_{\mathcal{F}(v)}(L_{v,b})\\ & \simeq \mathcal{F}(v\xrightarrow{a}e)(L_{v,a}[1-d(a,b)]) \simeq L_{e}[1-d(a,b)].\end{align*}

Since any equivalence, such as $T'[d(a,b)-1]$ , preserves derived endomorphisms, part (1) follows.