Braided nn-ary coherence conjecture for polyadic tensor categories

Let C\mathcal{C} be a braided semigroupal polyadic category with an nn-ary tensor product, an nn-ary associator \scalebox1.15\upshapeA(2n1)\scalebox{1.15}{\textsf{\upshape A}}^{\left(2n-1\right)\otimes}, and an nn-ary braiding \scalebox1.15\upshapeB(n)\scalebox{1.15}{\textsf{\upshape B}}^{\left(n\otimes\right)}. Assume that the associator satisfies the nn-ary coherence conditions giving the canonical reassociation isomorphism on 3n23n-2 objects, and that the braiding satisfies the polyadic analogue of the hexagon identity. Braided nn-ary coherence conjecture. Under these assumptions, every diagram containing \scalebox1.15\upshapeA(2n1)\scalebox{1.15}{\textsf{\upshape A}}^{\left(2n-1\right)\otimes} and \scalebox1.15\upshapeB(n)\scalebox{1.15}{\textsf{\upshape B}}^{\left(n\otimes\right)} commutes.

This is the braided extension of the proposed nn-ary associator coherence principle, analogous to the role of the hexagon axioms in ordinary braided monoidal categories. The source gives no evidence of a proof or refutation.

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Primary source

Steven Duplij, “Graded medial n-ary algebras and polyadic tensor categories”, arXiv:2001.04165 (2020).

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