Quotient Perverse Schober

• The paper demonstrates how quotient perverse schobers realize Verdier quotients by collapsing a subsurface and recovering global sections from a ribbon graph.
• The construction uses graded weighted decorated marked surfaces and mixed-angulations to encode geometric data and establish local spherical functor models.
• The theory links exchange graph combinatorics with stability conditions, providing a categorical bridge to quadratic differentials on collapsed surfaces.

A quotient perverse schober is, in the most precise current sense, a perverse-schober construction whose global sections realize the categorical effect of collapsing a subsurface and passing to a Verdier quotient. In "Categorical realization of collapsing subsurfaces and perverse schobers" (Fan et al., 30 Sep 2025), the construction starts from a graded weighted decorated marked surface $\mathbb S$, a subsurface $\Sigma\subset \mathbb S$, and the quotient triangulated category

$$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$

then produces a quotient perverse schober on the collapsed ribbon graph whose global sections recover this quotient. In the broader schober literature, however, quotient-language is heterogeneous: perverse schobers are generally categorifications of perverse sheaves, and their global categories are often obtained by gluing via limits rather than by quotienting in the Verdier sense (Kapranov et al., 2014, Christ et al., 2023).

1. Local categorical structure

The foundational one-point model of a perverse schober is a spherical functor. In the original formulation, a perverse schober on a disk with one singular point is given by an exact functor

$S:\mathcal{D}_0 \to \mathcal{D}_1$

with left and right adjoints and associated twist and cotwist functors; this is the categorical analogue of the quiver description of $\operatorname{Perv}(\Delta,0)$ (Kapranov et al., 2014). In stable $\infty$-categorical language, an adjunction

$F:\mathcal A \rightleftarrows \mathcal B:G$

is spherical when the twist and cotwist are equivalences, and the relative Waldhausen construction

$$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$

provides a local model for the stalks of a schober at a singularity; $S_1(F)$ is the section category of the Grothendieck construction of $F$ (Dyckerhoff et al., 2021).

For ribbon-graph models, the local singularity is a spider. A perverse schober on the $\Sigma\subset \mathbb S$0-spider is a spherical adjunction

$\Sigma\subset \mathbb S$1

while for $\Sigma\subset \mathbb S$2 it is a collection

$\Sigma\subset \mathbb S$3

satisfying

$\Sigma\subset \mathbb S$4

together with

$\Sigma\subset \mathbb S$5

Globalizing this, a parametrized perverse schober on a ribbon graph $\Sigma\subset \mathbb S$6 is a functor

$\Sigma\subset \mathbb S$7

whose restriction near each vertex is a perverse schober on the corresponding spider (Christ et al., 2023).

This local package is essential for quotient perverse schobers: the quotient construction does not replace the local spherical data, but reorganizes it around a collapsed vertex.

2. Surface-theoretic input and collapse

The geometric input in the quotient construction is a graded weighted decorated marked surface

$\Sigma\subset \mathbb S$8

where $\Sigma\subset \mathbb S$9 is a compact oriented surface with nonempty boundary, $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$0 is a finite nonempty set of marked points, $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$1 is a finite nonempty set of decorations, and $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$2 is a weight function (Fan et al., 30 Sep 2025). The grading is a foliation $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$3 on $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$4 satisfying

$$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$5

for each $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$6.

A mixed-angulation $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$7 is a finite set of graded open arcs cutting $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$8 into polygons such that a decoration $$D(\overline{\mathbb S}) \;:=\; D(\mathbb S)\big/ D(\Sigma),$$9 of weight $S:\mathcal{D}_0 \to \mathcal{D}_1$0 is enclosed in a $S:\mathcal{D}_0 \to \mathcal{D}_1$1-gon, and if $S:\mathcal{D}_0 \to \mathcal{D}_1$2 are successive edges around a polygon then

$S:\mathcal{D}_0 \to \mathcal{D}_1$3

Forward flips define the exchange graph $S:\mathcal{D}_0 \to \mathcal{D}_1$4 (Fan et al., 30 Sep 2025).

The collapse operation starts from a subsurface $S:\mathcal{D}_0 \to \mathcal{D}_1$5. If $S:\mathcal{D}_0 \to \mathcal{D}_1$6 is a boundary component of a connected component $S:\mathcal{D}_0 \to \mathcal{D}_1$7, then each $S:\mathcal{D}_0 \to \mathcal{D}_1$8 is required to be either internal to $S:\mathcal{D}_0 \to \mathcal{D}_1$9 or an actual boundary component of $\operatorname{Perv}(\Delta,0)$0, with the number of marked points on $\operatorname{Perv}(\Delta,0)$1 matching its winding number plus $\operatorname{Perv}(\Delta,0)$2. The collapsed surface $\operatorname{Perv}(\Delta,0)$3 is obtained by filling each $\operatorname{Perv}(\Delta,0)$4 with a disc carrying one new decoration of weight

$\operatorname{Perv}(\Delta,0)$5

The resulting slogan is that a subsurface is “shrunk to a point,” and its topology is converted into a weighted decorated singularity. The paper emphasizes that this setup includes simple poles, i.e. weight $\operatorname{Perv}(\Delta,0)$6, which were not treated in the earlier work of Barbieri–Möller–Qiu–So (Fan et al., 30 Sep 2025). This suggests that quotient perverse schobers are not merely abstract categorical quotients: they encode a geometric collapse in the same combinatorial language that already governs mixed-angulations and quadratic differentials.

3. Verdier quotient and the quotient schober

On the triangulated side, the quotient construction assumes a category $\operatorname{Perv}(\Delta,0)$7 attached to the surface and satisfying three CHQ-type hypotheses: every graded closed arc $\operatorname{Perv}(\Delta,0)$8 determines an object $\operatorname{Perv}(\Delta,0)$9; smoothing two closed arcs at an intersection corresponds to a cone; and the dual arcs of a mixed-angulation generate a finite heart $\infty$0 (Fan et al., 30 Sep 2025). Concretely, if $\infty$1 intersect at a decoration $\infty$2 with

$\infty$3

then there is a unique morphism

$\infty$4

whose cone is the object of the smoothed arc: $\infty$5

Let $\infty$6 be the thick subcategory generated by objects associated to arcs lying entirely inside $\infty$7. The quotient category is the Verdier quotient

$\infty$8

with short exact sequence

$\infty$9

The arc-to-object correspondence descends: for a graded closed arc $F:\mathcal A \rightleftarrows \mathcal B:G$0 on $F:\mathcal A \rightleftarrows \mathcal B:G$1, choose a lift $F:\mathcal A \rightleftarrows \mathcal B:G$2 upstairs and define

$F:\mathcal A \rightleftarrows \mathcal B:G$3

This is well-defined because arcs that become homotopic after collapse map to isomorphic objects in the quotient (Fan et al., 30 Sep 2025).

The perverse-schober realization uses a ribbon graph $F:\mathcal A \rightleftarrows \mathcal B:G$4, a connected subgraph $F:\mathcal A \rightleftarrows \mathcal B:G$5 corresponding to $F:\mathcal A \rightleftarrows \mathcal B:G$6, and a $F:\mathcal A \rightleftarrows \mathcal B:G$7-parametrized perverse schober

$F:\mathcal A \rightleftarrows \mathcal B:G$8

Its global sections are

$F:\mathcal A \rightleftarrows \mathcal B:G$9

After collapsing $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$0 to a single vertex $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$1, producing $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$2, the restriction of $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$3 to $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$4 gives a schober $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$5, and the new local data at $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$6 are built from adjunctions

$$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$7

The paper verifies that these adjunctions satisfy the perverse-schober axioms. It then introduces a degenerate schober $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$8 supported on the collapsed part, with zero adjunctions, and identifies the quotient schober with the orthogonal complement. The resulting global sections fit into a cofiber sequence of stable $$S_n(F)=S_{n+1}\mathcal B\times_{S_n\mathcal B}S_n\mathcal A$$9-categories

$S_1(F)$0

and hence a short exact sequence on homotopy categories

$S_1(F)$1

This is the schober-theoretic realization of the Verdier quotient (Fan et al., 30 Sep 2025).

The distinctive feature of the quotient perverse schober is therefore exact: it packages a Verdier quotient as a local-to-global construction on a collapsed ribbon graph.

4. Hearts of quotient type, flips, and exchange graphs

The quotient category carries a distinguished class of hearts. If $S_1(F)$2 is the heart attached to a refinement $S_1(F)$3 of a mixed-angulation on the collapsed surface, then

$S_1(F)$4

is a Serre subcategory, and the quotient

$S_1(F)$5

is a heart in $S_1(F)$6. These are the hearts of quotient type (Fan et al., 30 Sep 2025).

The principal combinatorial-categorical theorem is the identification

$S_1(F)$7

More precisely, the principal part of the exchange graph of mixed-angulations on the collapsed surface is isomorphic to the principal part of the exchange graph of hearts in the quotient category, compatibly with forward and backward flips and simple tilting (Fan et al., 30 Sep 2025).

This quotient theorem extends a pattern already established for noncollapsed schober categories. In the surface-schober framework of (Christ et al., 2023), positivity of the arc system kit implies that the edge-objects form a simple-minded collection and hence a finite heart, with

$S_1(F)$8

In the same framework, flipping an edge in the $S_1(F)$9-graph corresponds to simple tilting of the heart at the corresponding simple object, and the backward tilt is described by

$F$0

The quotient construction on collapsed surfaces preserves this flip/tilt paradigm, but the tilt now occurs after passing through the Verdier quotient (Christ et al., 2023).

A key technical step is that a flip on the collapsed surface can be lifted to a finite sequence of flips on a refinement upstairs. This is what makes the quotient exchange graph agree with the combinatorics of the collapsed surface (Fan et al., 30 Sep 2025).

5. Other meanings of “quotient” in the schober literature

The phrase “quotient perverse schober” should be distinguished from several earlier quotient-like uses of schober formalism.

Context	Mechanism
Collapsing subsurfaces	Verdier quotient realized by a quotient schober (Fan et al., 30 Sep 2025)
Surface schobers and global sections	Limit/gluing, not Verdier quotient (Christ et al., 2023)
Ginzburg-algebra reconstruction	Homotopy colimit/pushout of local dg-categories (Christ, 2021)
Toric mutation wall-crossing	Inertia category of clean objects (Nadler, 2018)
GIT and quotient-stack constructions	Subcategories of $F$1, GIT quotients, factorization quotients (Špenko et al., 2019, Koseki et al., 2022)

In the weighted-surface theory of (Christ et al., 2023), the global section category is

$F$2

equivalently the $F$3-category of coCartesian sections. The paper explicitly states that global sections are not presented as a quotient in the usual Verdier sense; the fundamental mechanism is gluing via a limit. Quotient-flavored features do appear locally—semiorthogonal decompositions, mutations, contractions of ribbon graphs, and models such as $F$4 with restriction $F$5—but the basic construction remains a glued section category rather than a quotient (Christ et al., 2023).

The Ginzburg-algebra paper gives a parallel local-to-global picture. For an ideal triangulation $F$6, there is a $F$7-parametrized perverse schober $F$8 with

$F$9

and an interior-supported version

$\Sigma\subset \mathbb S$00

Here the global category is reconstructed from local disc categories by homotopy colimits and pushouts, so the quotient-like aspect is descent and support, not Verdier quotienting (Christ, 2021).

In the toric-mutation setting, a schober on the sphere or cylinder yields two embeddings of the inertia category of clean objects into the global sections, and their comparison produces the wall-crossing transformation

$\Sigma\subset \mathbb S$01

This passage to inertia is quotient-like in the sense of objects equipped with monodromy automorphisms, but it is not a quotient schober in the collapsed-surface sense (Nadler, 2018).

Finally, several schober constructions arise from quotient geometry rather than quotient categories. The GIT schober of (Špenko et al., 2019) extends the Halpern-Leistner–Sam local system across a hyperplane arrangement using subcategories of $\Sigma\subset \mathbb S$02, while (Koseki et al., 2022) uses the derived factorization quotient

$\Sigma\subset \mathbb S$03

and VGIT windows in the construction of a schober on $\Sigma\subset \mathbb S$04. These are schobers built from quotient stacks or quotient categories, but not quotient perverse schobers in the specific Verdier-quotient-by-collapse sense.

6. Stability conditions and quadratic differentials

The principal application of the quotient perverse schober is the description of stability conditions by quadratic differentials on the collapsed surface. For a quadratic differential $\Sigma\subset \mathbb S$05 on a Riemann surface $\Sigma\subset \mathbb S$06, the relevant local coordinates are periods on the anti-invariant hat-homology

$\Sigma\subset \mathbb S$07

and on the saddle-free locus the stability function is

$\Sigma\subset \mathbb S$08

where $\Sigma\subset \mathbb S$09 is the closed arc corresponding to the simple object $\Sigma\subset \mathbb S$10. The main theorem is an isomorphism of complex manifolds

$\Sigma\subset \mathbb S$11

extended from the saddle-free locus by the stratification by saddle and recurrent trajectories together with the “walls have ends” lemma (Fan et al., 30 Sep 2025).

This quotient theorem sits naturally beside the noncollapsed correspondence of (Christ et al., 2023), where

$\Sigma\subset \mathbb S$12

is a biholomorphism onto a union of connected components, and

$\Sigma\subset \mathbb S$13

is an isomorphism onto its image. That framework already includes weighted marked surfaces with zeros, poles, boundary singularities, and exponential singularities, with local models such as an $\Sigma\subset \mathbb S$14-gon for a zero of order $\Sigma\subset \mathbb S$15, a $\Sigma\subset \mathbb S$16-gon for a simple pole, and $\Sigma\subset \mathbb S$17 distinct $\Sigma\subset \mathbb S$18-gons for an exponential singularity of order $\Sigma\subset \mathbb S$19 (Christ et al., 2023).

The quotient perverse schober adds a further operation to this dictionary: collapsing a subsurface, realizing the resulting category as a Verdier quotient, identifying its hearts with mixed-angulations of the collapsed surface, and then transporting the entire structure to the stability manifold. In that sense, it is a categorical device that converts geometric collapse into schober gluing, Verdier quotient, exchange-graph combinatorics, and finally a quadratic-differential description of stability conditions (Fan et al., 30 Sep 2025).