I watched a bunch of Tim Logvinenko's video, talking about generalized braid category and its representations. The origin of the story is trying to understand Caustic-Kamnitzer's construction, namely, what acts on the ambient space's coh category rather than the slice's.

I didn't understand CK's construction, so I probably should go back and read.

reading their old paper, almost 20 years old, https://arxiv.org/pdf/math/0701194

First a quick review of RT theory that assigns to tangle $T$ a linear map $V^{\otimes n} \to V^{\otimes m}$.

Then says what a weak categorification is, a bunch of categories with extra grading, that assigns to a tangle $T$ some functor $\Phi_T$, and blah.

Bernstein-I.Frenkel-Khovanov conjectured a weak categorification, and proved by Stroppel.