  We construct a categorification of the braid groups associated with Coxeter
groups inside the homotopy category of Soergel's bimodules. Classical actions
of braid groups on triangulated categories should come from an action of this
monoidal category. We construct representations of this monoidal category on
category O of a complex semi-simple Lie algebra and on constructible sheaves
over flag varieties. We also consider general constructions of
self-equivalences as reflections around another category.
