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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?

  1. 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?
  2. 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?)
  3. 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:

  1. ribbon fusion categories
  2. 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).

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Question 2: Given a pivotal braided category $\mathcal{C}$, there are 2 ways to endow $\mathcal{C}$ with twists under which $\mathcal{C}$ is a rigid balanced category. Conversely, given a rigid balanced category $\mathcal{C}$, there are 2 ways to endow $\mathcal{C}$ with a pivotal structure under which $\mathcal{C}$ is braided pivotal. These constructions are mutually inverse.

This is all explained in detail in Appendix A.2 of my recent article with Henriques and Tener http://arxiv.org/abs/1509.02937. (These results are not original, but we give a comprehensive treatment there.) Of interest to you may also be Section 2.3, which gives a synoptic chart of the types of tensor category.

Question 3: No, not necessarily. For example, the center $\mathcal{Z}(\mathcal{C})$ of any pivotal fusion category $\mathcal{C}$ is pivotal braided, but it is not ribbon unless $\mathcal{Z}(\mathcal{C})$ is spherical. In general, there is no reason for this to be the case. In fact for a semisimple category $\mathcal{C}$, pivotal braided is ribbon if and only if it is spherical. This is Proposition A.4 in our preprint.

Question 1: I suspect the answer is that there is not necessarily any interesting twist in only the presence of a pivotal structure. I don't have an example off the top of my head though. But when you also have a braiding, you're in business as explained in the answer to Question 2 above.

So to sum up the bit at the end, yes, ribbon fusion is the same as spherical braided fusion, because the category is semisimple.

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    $\begingroup$ Re Q2: I will look in your paper for precise statements. Perhaps you mean categories over $\mathbb C$ with simple unit? I would have thought that the space of ways to go between these structures was a torsor for the group of square roots of $1$ among (braided?) monoidal endomorphisms of identity functor, or something like that. Are you saying that such a choice always exists? Or that if one exists, then there are two of them? $\endgroup$ – Theo Johnson-Freyd Oct 29 '15 at 14:22
  • $\begingroup$ @TheoJohnson-Freyd, I believe it's just the choice of over-braiding or under-braiding. $\endgroup$ – Manuel Bärenz May 15 '18 at 9:55
  • $\begingroup$ Question 1: Is this maybe answered in Bruce Bartlett's excellent article Fusion categories via string diagrams? * A fusion category $\mathcal{C}$ has a monoidal endofunctor $\mathcal{T}$, the "pivotal functor". Its underlying functor is the identity of the category, the monoidal coherences are the interesting data. * A pivotal structure is then a monoidal iso from the identity functor (with trivial coherences) to $\mathcal{T}$. * In particular, this gives an automorphism of the identity functor. I have the feeling it's the twist, but can't prove it. $\endgroup$ – Manuel Bärenz May 15 '18 at 9:57

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