Balanced monoidal and homotopy symmetric

Question

It's probably a very simple question but I am not sure about the reference.

In the definition of a balanced monoidal category we require that the braiding isomorphims $$c_{V, W}: V \otimes W \to W \otimes V$$ are not arbitrary but admit twisting isomorphisms $\theta_V: V \to V$ which, in a way, represent $c_{V, W} c_{W, V}$ as a coboundary.

Does it mean that from the homotopy point of view, existence of a twisting makes the category symmetric monoidal in the sense of $\infty$-categories?

Answer 1

No, a balanced monoidal structure is something different. One way to think about these things is in terms of the relevant operads. Braided monoidal means an algebra over the $E_2$ operad, while symmetric monoidal means an algebra over the $E_{\infty}$ operad.

Balanced monoidal means an algebra over the framed $E_2$ operad; basically the difference between this operad and the ordinary $E_2$ operad is that you are allowed to rotate the disks.