The NilHecke schober

Perverse schobers can be used to describe Fukaya categories but are hard to axiomatize and construct. In this paper, we give an explicit construction of a perverse schober intended to accurately describe the Fukaya category of the horizontal Hilbert scheme considered by Aganagic et al. in the frame of the knot invariant categorification program. The formalism is based on the notion of $A_n$-schobers of Dyckerhoff and Wedrich. The construction is entirely algebraic and we check the schober axioms with the help of diagrammatic calculus.