Cyclic structure on a balanced (or ribbon) monoidal category

Question

As it is well known, a balanced (and in particular ribbon) monoidal category is an algebra over the framed little 2-discs operad. The latter is homotopy equivalent to the operad of moduli space of genus 0 surfaces with boundaries, hence has a natural cyclic structure.

What additional structure correspond to a cyclic algebra in categories over this operad ?

Usually, in vector spaces, a cyclic algebra over a cyclic operad has a non-degenerate pairing compatible with the operadic operations. One marvelous property of categories is that the already have a canonical pairing given by hom spaces, and under mild conditions it is non-degenerate. So I guess that a cyclic structure in that case should be a certain compatibility between the twist/ribbon element and the home spaces.

Answer 1

In https://arxiv.org/abs/2010.10229 we characterize cyclic framed little 2-disks algebras in any symmetric monoidal bicategory.

In the case that this symmetric monoidal bicategory is given by finite categories, left exact functors and their transformations, it turns out that they amount precisely to balanced Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld, i.e. braided monoidal categories with a weaker form a rigidity that also have a compatible balancing.

Note: Our article tries to shed some light on the relation of cyclicity of certain operads in low-dimensional topology to duality properties of representation categories appearing in quantum algebra, and your question was extremely helpful for that. Thank you!