Starting from braided monoidal categories of quantum group representations and categorification results one comes to the question whether categorification can be used to construct a corresponding braided monoidal 2-category. In this talk we start from the observation that the Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A. We then discuss the problem of categorification and present some answers using Soergel bimodule categories. This is on joint work with Aaron-Maazel Gee, Leon Liu, David Reutter and Paul Wedrich.
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