Perverse schobers of Coxeter type $\mathbb{A}$
Tobias Dyckerhoff, Paul Wedrich
TL;DR
The paper introduces and rigorously develops $oldsymbol{A}_n$-schobers as categorifications of perverse sheaves on symmetric products $ ext{Sym}^{n+1}(oldsymbol{oldsymbol{C}})$, and shows that each such schober produces a categorical action of the braid group $ ext{Br}_{n+1}$. It builds a central example from singular Soergel bimodules, establishing a categorified graded bialgebra structure governed by Rickard complexes and parabolic induction, and connects this to type-$ ext{A}$ Schur algebroids upon decategorification. The framework is organized through higher Beck–Chevalley cubes, with concrete low-rank instances (A1, A1×A1, A2) and a general An construction, culminating in a factorizing family of Soergel schobers that realize a rich, braided higher-categorical structure. Together, the results point toward a categorified graded bialgebra in a freely generated braided monoidal $( ext{∞},2)$-category, with strong links to perverse sheaves, parabolic induction, and link homology theories.
Abstract
We define the concept of an $\mathbb{A}_n$-schober as a categorification of classification data for perverse sheaves on $\mathrm{Sym}^{n+1}(\mathbb{C})$ due to Kapranov-Schechtman. We show that any $\mathbb{A}_n$-schober gives rise to a categorical action of the Artin braid group $\mathrm{Br}_{n+1}$ and demonstrate how this recovers familiar examples of such actions arising from Seidel-Thomas $\mathbb{A}_n$-configurations of spherical objects in categorical Picard-Lefschetz theory and Rickard complexes in link homology theory. As a key example, we use singular Soergel bimodules to construct a factorizing family of $\mathbb{A}_n$-schobers which we refer to as Soergel schobers. We expect such families to give rise to a categorical analog of a graded bialgebra valued in a suitably defined freely generated braided monoidal $(\infty,2)$-category.