Generalised braid category

I will describe a joint project with Rina Anno: a categorification of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. A decade old conjecture states that this category acts on the derived categories of (the cotangent bundles of) full and partial flag varieties.
The basic building blocks of this conjectural action are spherical, P- and (what we expect to be) Grassmanian functors together with the equivalences they induce. I will describe our present progress and future expectations.