Studying groups through their linear representations is a well established domain of research. We will be particularly interested in groups appearing in low-dimensional topology. Braids groups, for instance, are known to be linear since the work of Bigelow and Krammer.
The representation used to obtain this result goes through the action of braids on the homology of covering spaces of configuration spaces of at least two points in the punctured disc. The case of a single point corresponds to the classical Burau representation which is know to have kernel when the number of strands is at least five by work of Bigelow, Long, Moody and Paton. The case of four strands remains an open question. This very classical representation has been categorified by Khovanov and Seidel: one can explicitly describe a faithful categorical braid group action on the homotopy category of modules over a so called zig-zag algebra. Hence when categorifying one can gain faithfulness. In this talk, I will review these various constructions and explain the ideas behind the faithfulness results and how they interplay. This talk is based on joint work in progress with Hoel Queffelec and Anne-Laure Thiel. | 
