Hypergroup structure from the quotient by the local subcategory in a weak-braided monoidal category

Determine whether, for any weak-braided monoidal category D, defining the local subcategory D^{loc} as the full weak-braided subcategory equivalent to a braided monoidal category (interpreted as the kernel of the weak braiding), the quotient K = D // D^{loc} carries a canonical hypergroup structure, analogous to hypergroups arising from quotients in module categories over commutative algebra objects.

Background

The paper introduces weak-braided monoidal categories via an operadic approach (the operad wPaB) and shows that categories of modules over commutative algebra objects in braided monoidal categories, as well as braided G-crossed categories, are weak-braided. In the context of extensions by algebra objects, prior work exhibits a hypergroup structure on a quotient of module categories by their local (braided) subcategory.

Motivated by this, the author proposes defining a local subcategory D^{loc} inside a general weak-braided category D as the kernel of the weak braiding (a weak-braided subcategory equivalent to a braided category). The question is whether the quotient K = D // D^{loc} naturally inherits a hypergroup structure in this broader, non-invertible symmetry setting.