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I know that it has been shown that $E_2$ algebra objects in Categories are simply braided monoidal categories. In particular, Lurie says that an $E_2$-monoidal structure on the infinity-category $N(C)$ (the nerve of a category C) is a braided monoidal structure on C.

Is there a generalization (not sure if this is the right word) of this where instead of having braiding isomorphisms we have: $b_{V,W}:V\otimes W\to W\otimes V$ is a quasi-isomorphism? Obviously, we need some dg-structure, so we would be talking about an $E_2$-monoidal structure on the dg-nerve of a dg-category.

What would the implications be for the composition $b_{W,V}\circ b_{V,W}$ when compared to the identity quasi-isomorphism $\mathrm{id}:V\otimes W\to V\otimes W$?

I am interested in possible new knot invariants.

Any thoughts or improvements on how I am asking the question are highly appreciated.

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    $\begingroup$ Any generalization will require you to choose a cellular model for the $E_2$ operad. I.e. there wont be anything as simple as what you suggest-- in this generality you need a lot more data than just the braiding maps. You could use the nerve of the groupoid of parenthesized braids, or Batanin's work (related to the category Theta_2). $\endgroup$
    – Phil Tosteson
    Jan 8, 2020 at 20:30
  • $\begingroup$ What if I choose the brace operad. So there are 2 types of tensors (left/right multiplication and up/down multiplication). I realize the generality that I want requires a lot of data, I'm trying to see if it is managable in some sort of "small" model version of E2. $\endgroup$
    – Matthew Levy
    Jan 8, 2020 at 21:25

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