A bimodule description of the Hecke category

For a Coxeter system and a representation V of this Coxeter system, Soergel defined a category which is now called the category of Soergel bimodules and proved that this gives a categorification of the Hecke algebra when V is reflection faithful. Elias and Williamson defined another category when V is not reflection faithful and proved that this category is equivalent to the category of Soergel bimodules when V is reflection faithful. Moreover, they proved the categorification theorem for their category with fewer assumptions on V . In this paper, we give a bimodule description of the Elias–Williamson category and re-prove the categorification theorem.