Let $\mathcal{C}$ be a pivotal tensor category. Feel free to assume finiteness, semisimplicity, fusion, sphericality, unitarity or whatever makes things interesting. Which of the following structures come for free?

- Does the pivotal structure canonically define a
*twist*: a natural transformation $\theta$ from the identity functor to itself? Or must I to add this structure by hand? - Endow $\mathcal{C}$ with a braided structure $c$ (assume this is possible). Is $\mathcal{C}$
*balanced*: do $\theta$ and $c$ satisfy $\theta_{x\otimes y}=c_{yx}\cdot c_{xy}\cdot(\theta_y\otimes\theta_x)$? (*ie*does the ribbon diagram make sense?) - If so, is $\mathcal{C}$
*ribbon*: does $\theta$ satisfy $(\theta_x)^*=\theta_{x^*}$? (compatible rigidity, twist, and braiding)

I'm asking because *premodular* tensor categories appear in the literature with (at least) two definitions:

- ribbon fusion categories
- spherical braided fusion categories

Ignoring the absence of sphericality from the first definition (is it implied by ribbon?), these definitions disagree unless the answers to the above are affirmative (perhaps assuming fusion and sphericality).