Equivariantization-like reconstruction for weak-braided categories

Develop an equivariantization-style reconstruction procedure for weak-braided monoidal categories that, given the weak-braided category Rep_C A of A-modules in a braided monoidal category C, recovers C in analogy with the equivariantization procedure in the braided G-crossed setting.

Background

In the braided G-crossed case, there is a well-established equivariantization procedure that reconstructs the original braided category from its G-crossed data and module categories.

The paper asks whether a similar reconstruction mechanism exists in the more general weak-braided context, where the symmetry need not be group-like, thereby potentially providing a method to recover C from Rep_C A when only a weak-braided structure is available.