I will recall Lusztig's Jcategories, which are fusion categories obtained from the perverse homology of Soergel bimodules. For example, for finite dihedral Coxeter type this fusion category is the semisimplified quotient of the module category of quantum so(3) at a root of unity, which ReshetikhinTuraev and TuraevViro used for the construction of 3dimensional Topological Quantum Field Theories. In the second part of my talk, I will recall the basics of 2representation theory, which is the categorical analogue of representation theory, and indicate how the fusion categories above can conjecturally be used to study the 2representation theory of Soergel bimodules of finite Coxeter type. This is joint work with Mazorchuk, Miemietz, Tubbenhauer and Zhang.
