Perverse schobers of Coxeter type $\mathbb{A}$

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.