  We characterize cyclic algebras over the associative and the framed little
2-disks operad in any symmetric monoidal bicategory. The cyclicity is
appropriately treated in a coherent way, i.e. up to coherent isomorphism. When
the symmetric monoidal bicategory is specified to be a certain symmetric
monoidal bicategory of linear categories subject to finiteness conditions, we
prove that cyclic associative and cyclic framed little 2-disks algebras,
respectively, are equivalent to pivotal Grothendieck-Verdier categories and
ribbon Grothendieck-Verdier categories, a type of category that was introduced
by Boyarchenko-Drinfeld based on Barr's notion of a $\star$-autonomous
category. We use these results and Costello's modular envelope construction to
obtain two applications to quantum topology: I) We extract a consistent system
of handlebody group representations from any ribbon Grothendieck-Verdier
category inside a certain symmetric monoidal bicategory of linear categories
and show that this generalizes the handlebody part of Lyubashenko's mapping
class group representations. II) We establish a Grothendieck-Verdier duality
for the category extracted from a modular functor by evaluation on the circle
(without any assumption on semisimplicity), thereby generalizing results of
Tillmann and Bakalov-Kirillov.
