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?



Balanced monoidal and homotopy symmetric
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](https://ncatlab.org/nlab/show/framed+little+2-disk+operad); basically the difference between this operad and the ordinary $E\_2$ operad is that you are allowed to rotate the disks. 


