Soergel Calculus with patches

Abstract

We study morphism complexes between dg lifts of Rouquier complexes. We recover the transitive system of homotopy equivalences between different lifts corresponding to the same braid, as well as the Rouquier formula. We develop a large scale version of Gaussian elimination for complexes and use it to reduce morphism spaces of the form Hom • (1, Fw) where Fw is the Rouquier complex associated with a Coxeter word w.