Perverse Schobers: Categorical Monodromy

• Perverse schobers are categorical analogs of perverse sheaves that replace vector spaces with triangulated, dg-, or stable ∞-categories, capturing both local and global monodromy data.
• They are constructed via spherical functors, spherical pairs, and diagrammatic models on disks, hyperplane arrangements, surfaces, and symmetric powers to encode geometric transitions.
• Applications span birational flops, conifold degenerations, mirror symmetry, and categorical braid group actions, making them a versatile tool in modern algebraic geometry.

Searching arXiv for recent and foundational papers on perverse schobers. Perverse schobers are categorical analogs of perverse sheaves: they replace the vector spaces and linear maps in the local and global description of perverse sheaves by triangulated, dg-, or stable $\infty$-categories together with exact functors, adjunctions, and autoequivalences. In the foundational local model on a disk with one singular point, the relevant datum is a spherical functor; in more elaborate settings, one encounters spherical pairs, diagrams over hyperplane arrangements, ribbon-graph-parametrized schobers on surfaces, and Coxeter-type schobers on symmetric powers. Across these formalisms, schobers organize categorical monodromy, nearby and vanishing cycle phenomena, wall-crossing, and birational or symplectic transitions, with applications ranging from flops and homological mirror symmetry to conifold degenerations, stability conditions, cluster structures, and categorical braid group actions (Kapranov et al., 2014).

1. Local models and foundational definitions

The basic local model of a perverse schober is the categorification of a perverse sheaf on a disk $\Delta$ with one critical point. In the Kapranov–Schechtman framework, this is encoded by a spherical functor

$F:\mathcal{A}\to\mathcal{B}$

with left and right adjoints $F^L$ and $F^R$, together with the twist and cotwist

$$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$

required to be autoequivalences. In this model, $\mathcal{B}$ plays the role of the generic fiber category, $\mathcal{A}$ the singular sector, and the spherical twist categorifies monodromy around the puncture (Kapranov et al., 2014).

A symmetric reformulation uses spherical pairs. Here one starts with a triangulated category equipped with two admissible subcategories and semiorthogonal decompositions of the form

$$\mathcal{E}_0=\langle \mathcal{E}_-^\perp,\mathcal{E}_-\rangle =\langle \mathcal{E}_+^\perp,\mathcal{E}_+\rangle,$$

together with equivalences between the corresponding components. Spherical pairs produce spherical functors and furnish a formulation that is particularly adapted to wall-crossing and flop geometry (Kapranov et al., 2014).

Several later papers retain this local spherical-functor paradigm while changing the ambient categorical setting. In stable $\infty$-categorical language, a schober on $\Delta$0 with a singularity at $\Delta$1 is again described by a spherical adjunction, and the resulting 2-category of spherical functors admits a spectral description as modules over a convolution category of coherent sheaves on a correspondence built from $\Delta$2 and $\Delta$3 (Gammage et al., 2022). This suggests that the local theory of schobers sits naturally at the intersection of homological algebra, microlocal geometry, and higher categorical representation theory.

A persistent foundational point is that “perverse schober” is not a single universal formalism. Some papers describe schobers as conjectural categorical analogs of perverse sheaves, whereas others give precise models on particular bases—disks, hyperplane arrangements, ribbon graphs, or symmetric powers—using spherical adjunctions, Beck–Chevalley cubes, or local systems of categories (Kapranov et al., 2014). This is not a contradiction: it reflects the fact that the foundational local picture is rigid, while global axiomatizations vary with the geometric context.

2. Global formalisms on curves, arrangements, and surfaces

On a disk with several marked points, a schober is organized by a skeleton of arcs from a boundary basepoint to the singularities. Changing the skeleton is controlled by the braid group, and the corresponding transformation of local spherical-functor data gives the categorical counterpart of the braid action on quiver descriptions of perverse sheaves (Kapranov et al., 2014). On a general Riemann surface, one glues a disk schober near the singular locus to a local system of categories on the complement; in this way schobers on curves encode both local singularity data and global monodromy (Donovan, 2018).

For real or complexified hyperplane arrangements, Kapranov–Schechtman’s combinatorial description of perverse sheaves has a categorical analog in the form of $\Delta$4-schobers. Here one assigns a triangulated category to each face and adjoint functors to each incidence, subject to axioms of idempotency, invertibility across facets, and collinear transitivity. In the GIT setting of quasi-symmetric representations, such an $\Delta$5-schober extends Halpern-Leistner–Sam’s local system on the stringy Kähler moduli space and is built from explicit subcategories of $\Delta$6 generated by $\Delta$7-equivariant vector bundles (Špenko et al., 2019). This arrangement-theoretic formalism also underlies the categorification of quasi-symmetric GKZ systems, where specializing the decategorified schober recovers the associated non-resonant GKZ perverse sheaf and its monodromy (Špenko et al., 2020).

A different but related globalization uses ribbon graphs and exit-path categories. A $\Delta$8-parametrized perverse schober on a ribbon graph $\Delta$9 is a functor

$F:\mathcal{A}\to\mathcal{B}$0

whose restriction to the star of each vertex is locally perverse in the sense of spherical adjunctions. The global sections are defined as a limit and can be interpreted as coCartesian sections of the associated Grothendieck construction (Christ et al., 2023). This language is particularly effective on surfaces with boundary, where schobers on spanning graphs provide categories of global sections that behave like topological Fukaya categories with coefficients.

The same ribbon-graph formalism supports explicit induction and restriction functors between local and global sections. In marked surfaces with boundary, evaluation on vertices or edges admits adjoints constructed by gluing along clockwise and counterclockwise trajectories, and the resulting global boundary restriction is spherical. This produces Frobenius exact structures on global sections and allows cluster tilting subcategories to be glued from local vertex data (Christ, 1 Sep 2025). In this sense, surface schobers do not merely record local monodromy; they also provide a mechanism for assembling exact and cluster-theoretic structures globally.

A further extension appears in the study of collapsed surfaces and Verdier quotients. There, quotient perverse schobers on collapsed ribbon graphs produce cofiber sequences of global sections that recover Verdier quotient sequences of triangulated categories, and the exchange graph of hearts in the quotient category is identified with the exchange graph of mixed-angulations on the collapsed surface (Fan et al., 30 Sep 2025). This shows that schober global sections can realize nontrivial quotient constructions, not only local-to-global gluings.

3. Birational geometry, GIT, and mirror symmetry

One of the most developed sources of perverse schobers is birational geometry. For simple flops of relative dimension $F:\mathcal{A}\to\mathcal{B}$1, the derived categories of the two small resolutions and of their fiber product form the categorical analog of the hyperbolic-stalk diagram of a perverse sheaf. The half-monodromies are the classical flop functors

$F:\mathcal{A}\to\mathcal{B}$2

and the resulting “flober” admits categorical cohomology objects whose $F:\mathcal{A}\to\mathcal{B}$3 and compactly supported $F:\mathcal{A}\to\mathcal{B}$4 identify with $F:\mathcal{A}\to\mathcal{B}$5 and $F:\mathcal{A}\to\mathcal{B}$6 for the singular target $F:\mathcal{A}\to\mathcal{B}$7 (Bondal et al., 2018). This makes the analogy with perverse sheaf cohomology literal at the categorical level.

Donovan’s construction of schobers on Riemann surfaces makes this birational picture global. For simple balanced toric GIT wall crossings, one obtains a schober on $F:\mathcal{A}\to\mathcal{B}$8 with singularities at $F:\mathcal{A}\to\mathcal{B}$9 whose generic fiber is the derived category of one GIT quotient and whose local monodromies are the spherical twists attached to window-shift equivalences. In standard flop situations, this specializes to a schober with generic fiber $F^L$0 and monodromy given by the spherical twist around $F^L$1 (Donovan, 2018). In the threefold case, this is related to a further schober on a partial compactification of the stringy Kähler moduli space.

The Grothendieck resolution furnishes a more intricate “web of flops.” In that setting, the cells of the coroot hyperplane arrangement index birational models, and the associated derived categories and pull-push functors form an $F^L$2-schober-type diagram. For $F^L$3, the central object is related to Schubert’s classical variety of complete triangles, and most of the categorical arrangement axioms are verified in this example (Bondal et al., 2018). This shows that schobers can encode not only isolated wall crossings but also higher-dimensional wall-crossing networks.

Mirror symmetry sharpens this picture. For flops in dimensions $F^L$4 and $F^L$5, mirror symmetry for perverse schobers is established by identifying the coherent-side spherical-pair schober with a symplectic-side schober built from partially wrapped Fukaya categories with stops determined by FLTZ skeleta. In the Atiyah flop and $F^L$6 surface singularity cases, one has equivalences of schobers

$F^L$7

and the spherical-pair functors on the coherent side match stop-inclusion and stop-removal functors on the Fukaya side (Donovan et al., 2019). The coherent–constructible correspondence and the Ganatra–Pardon–Shende equivalence provide the bridge through wrapped constructible sheaves.

A parallel but distinct mirror-symmetric construction appears for Calabi–Yau hypersurfaces. There, a schober on $F^L$8 categorifies the intersection complex of the mirror-symmetry local system. The local monodromies are realized as twists or dual twists of spherical functors at the large complex structure point, the conifold point, and the orbifold point, and Orlov’s equivalence between graded matrix factorizations and $F^L$9 supplies the transition maps (Koseki et al., 2022). This demonstrates that schobers can capture categorical monodromy in moduli problems beyond birational wall crossings.

These examples collectively show that schobers are not merely formal categorifications: they are effective devices for packaging derived equivalences, spherical twists, and mirror correspondences into a geometric object over a base.

4. Conifold degenerations, nearby cycles, and finite-node structures

Perverse schobers also arise in degenerations, especially conifold degenerations of Calabi–Yau threefolds. In a one-parameter degeneration $F^R$0 with a single ordinary double point in the central fiber, nearby and vanishing cycles of

$F^R$1

lie in $F^R$2, and the variation morphism

$F^R$3

produces a canonical perverse object

$F^R$4

Under the hypothesis that $F^R$5 is a monomorphism, this object restricts to $F^R$6 on the smooth locus and fits into a short exact sequence

$F^R$7

so it differs from the intersection complex by a single rank-one contribution supported at the node (Rahman, 1 Apr 2026).

This construction is explicitly tied to the categorical picture. In the local Kapranov–Schechtman model, a schober on the disk consists of a bulk category, a singular sector, and a spherical functor whose twist gives the categorical monodromy. Decategorification via $F^R$8 and a realization map recovers the Picard–Lefschetz operator

$F^R$9

on $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$0, where $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$1 is the vanishing cycle. The perverse object $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$2 is then the sheaf-theoretic shadow of the same rank-one spherical monodromy phenomenon (Rahman, 1 Apr 2026).

For finitely many nodes, the local ordinary-double-point blocks assemble into a finite-node schober datum with one localized categorical sector per node and a common bulk category. Its decategorified shadow is the corrected finite-node perverse extension

$$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$3

and this is compatible with the mixed-Hodge-module refinement

$$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$4

(Rahman, 8 Apr 2026). The resulting “bulk/localized-sector” architecture is a theorem-level categorical model for finite-node conifold degenerations.

A notable feature of this finite-node formalism is its quiver shadow: one bulk vertex and one localized vertex for each node, with arrows induced by the attachment functors. This provides a first precursor of a wall-crossing formalism in which the localized sectors move and recombine as the degeneration data vary (Rahman, 8 Apr 2026).

These degeneration-theoretic constructions clarify a frequent misconception. A perverse schober is not simply “a category with monodromy.” In the conifold setting, the essential datum is the relation between the bulk and the localized vanishing-cycle sectors, together with the way their decategorification reproduces nearby/vanishing cycle exact triangles and corrected perverse extensions.

5. Surface models, global sections, and stability conditions

On surfaces, perverse schobers provide a direct route from combinatorics of arcs and polygons to triangulated categories and stability spaces. Given a weighted marked surface with a mixed-angulation, one constructs a ribbon graph and a graph-parametrized schober whose global sections form a stable category. Under a positivity hypothesis on an arc system kit, the objects associated to the edges form a simple-minded collection and therefore determine a finite-length heart (Christ et al., 2023).

The central result is that mixed-angulations and flips correspond to hearts and simple tilts. A backward flip at an edge induces an equivalence of global-sections categories taking the canonical heart on the flipped graph to the backward tilt at the corresponding simple object (Christ et al., 2023). This identifies the exchange graph of mixed-angulations with a union of connected components in the exchange graph of finite hearts.

The geometric side of this correspondence is the moduli of framed quadratic differentials. For a weighted marked surface $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$5, the period map

$$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$6

gives local coordinates on the framed moduli space, and the central charge of the corresponding stability condition is obtained from the same period integral under the identification of $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$7 with the relevant twisted homology group (Christ et al., 2023). Under the schober hypotheses, this yields a biholomorphism from the moduli space of framed quadratic differentials onto a union of connected components of the stability space.

The same strategy extends to more singular differentials. Relative graded Brauer graph algebras were discovered from two routes: cyclic quotients of derived categories of relative Ginzburg algebras appearing as local schober stalks, and deformations of partially wrapped Fukaya categories of surfaces. Their derived categories are recovered as global sections of a surface schober, and the stability spaces of these categories are again described in terms of quadratic differentials (Christ et al., 2024). This generalizes the earlier picture from 3-Calabi–Yau triangulated-surface categories to a wider relative setting.

Schobers on ribbon graphs also realize Ginzburg algebras of triangulated surfaces. A $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$8-parametrized schober on the dual ribbon graph of an ideal triangulation has global sections equivalent to the unbounded derived category of the associated relative surface Ginzburg algebra, while sections supported on the interior recover the non-relative surface Ginzburg algebra (Christ, 2021). Flip equivalences between triangulations become equivalences of global sections produced by local modifications of the schober, yielding a new proof of derived invariance under mutation.

A closely related quotient construction appears for collapsed surfaces. There, a quotient perverse schober on the collapsed graph has global sections fitting into a cofiber sequence with the original global sections and the sections over the collapsed subsurface, thereby realizing the Verdier quotient category associated to the collapse (Fan et al., 30 Sep 2025). This suggests that schober global sections are a natural environment for relative and quotient phenomena in surface-type categories.

6. Coxeter type $$T_{\mathcal{B}}:=\operatorname{Cone}\bigl(F\circ F^R \to \operatorname{Id}_{\mathcal{B}}\bigr),\qquad C_{\mathcal{A}}:=\operatorname{Cone}\bigl(\operatorname{Id}_{\mathcal{A}} \to F^R\circ F\bigr)[-1],$$9, braid actions, and explicit algebraic models

A recent direction replaces ribbon graphs or disks by the stratification of $\mathcal{B}$0. An $\mathcal{B}$1-schober is a coherent diagram indexed by compositions of $\mathcal{B}$2 and subject to five axioms: adjunctability, recursiveness, far-commutativity, twist invertibility, and defect vanishing (Dyckerhoff et al., 11 Apr 2025). Higher twists are defined as total fibers of Beck–Chevalley cubes associated to bifactorization cubes, and they categorify the graded-bialgebra relations that classify perverse sheaves on $\mathcal{B}$3.

Any $\mathcal{B}$4-schober gives rise to a categorical action of the Artin braid group $\mathcal{B}$5 on the category over the open stratum. In the $\mathcal{B}$6 case this is the relation

$\mathcal{B}$7

derived from the biCartesian nature of the relevant Beck–Chevalley squares (Dyckerhoff et al., 11 Apr 2025). This recovers familiar braid actions from two sources emphasized in the paper: Seidel–Thomas $\mathcal{B}$8-configurations of spherical objects and Rickard complexes in link homology.

Singular Soergel bimodules provide a key example. The corresponding Soergel schobers form a factorizing family of framed $\mathcal{B}$9-schobers, and their higher twists are identified with graded shifts of Rickard crossing complexes (Dyckerhoff et al., 11 Apr 2025). This realizes link-homological braid actions as schober monodromy and suggests a categorical analog of a graded bialgebra valued in a freely generated braided monoidal $\mathcal{A}$0-category.

The NilHecke schober gives an explicit algebraic construction in the same Coxeter-type framework. It assigns to each composition $\mathcal{A}$1 of $\mathcal{A}$2 the category $\mathcal{A}$3 of perfect complexes over an appropriate product NilHecke algebra, and to each refinement the corresponding restriction functor with right adjoint induction. Its higher twists are fibers of explicitly constructed Beck–Chevalley cubes, and the full $\mathcal{A}$4-schober axioms are verified by diagrammatic NilHecke calculus (Kreeke, 29 Aug 2025). This construction is intended to model the Fukaya category of the horizontal Hilbert scheme studied in knot categorification.

A different “real analogue” of the schober idea appears in microlocal sheaf theory for Legendrian fronts. Under a purity condition requiring microstalks to be concentrated in degree $\mathcal{A}$5, one obtains local linear models across points and crossings, categorified by semiorthogonal decompositions and mutations. In this framework, spherical functors, $\mathcal{A}$6-spherical functors, mutation braiding, and irregular schobers arise from the same geometric picture (Kuwagaki, 2019). This does not replace the complex-curve formalism; rather, it extends the schober heuristic to a microlocal and irregular setting.

Taken together, Coxeter-type, NilHecke, Soergel, and Legendrian constructions show that perverse schobers now occupy a substantial part of the interface between braid groups, categorical representation theory, link homology, and symplectic topology. A plausible implication is that the subject’s long-standing foundational ambiguity is being resolved not by a single universal definition, but by several precise and interoperable formalisms tailored to the geometry of the base.