A Coxeter category is a braided tensor category which carries an action of a generalised braid group $B_W$ on its objects. The axiomatic of a Coxeter category and the data defining the action of $B_W$ are similar in flavor to the associativity and commutativity constraints in a monoidal category, but are related to the coherence of a family of fiber functors.
We will show how to construct two examples of such structure on the integrable category $\mathcal O$ representations of a symmetrisable Kac–Moody algebra $\mathfrak g$, the first one arising from the quantum group $U_\hbar \mathfrak g$, and the second one encoding the monodromy of the $KZ$ and Casimir connections of $\mathfrak g$.
The rigidity of this structure, proved in the framework of PROP categories, implies in particular that the monodromy of the latter connection is given by the quantum Weyl group operators of $U_\hbar \mathfrak g$.
This is a joint work with Valerio Toledano Laredo.