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I understood more or less the claim that $k$-tuply monoidal $n$-categories can be seen as $n$-categories equipped with an action of the $E_k$-operad.

For $k=2$, we have a homotopy equivalence $E_2(r) \cong K(P_r, 1)$, where $P_r$ is the pure braid group on $r$ strands. Let $\mathcal{C}$ be a braided monoidal category, so $E_2$ acts on $\mathcal{C}$. Moreover, we have an action of $P_r$ on objects of the form $V_1 \otimes \dots \otimes V_r \in \mathcal{C}$.

My question is, is there some higher analogue to $P_r$ for $k>2$?

Edit: the answer is basically given in the comments of this question: Braid 2-groups, symmetric 2-groups

However, I would be very happy about a definitive answer in terms of generators and relations of this 2-group. As far as I understood, it should involve the Zamolodchikov Tetrahedron equation.

Edit: I just noticed that I'm actually also interested in $n > 1$ (not just higher $k$), which my previous Edit refers to.

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    $\begingroup$ Yes. This is worked out in my paper with Michael Batanin "Homotopy theory of algebras of substitudes and their localisation". Search for "higher braided operad" in that paper. ams.org/journals/tran/2022-375-05/S0002-9947-2022-08600-1/… $\endgroup$
    – David White
    Commented Jul 15 at 13:00
  • $\begingroup$ Thx, I'll have a look! $\endgroup$
    – Bipolar Minds
    Commented Jul 15 at 13:09
  • $\begingroup$ Could you maybe be a bit more specific, where the desired objects come up in your paper and how? $\endgroup$
    – Bipolar Minds
    Commented Jul 15 at 13:37
  • $\begingroup$ You can email me. $\endgroup$
    – David White
    Commented Jul 15 at 15:38

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