Modular functors are collections of projective representations of mapping class groups of surfaces, compatible with cutting and gluing operations. They can be thought of as categorified, anomalous 2d topological field theories (TFT) where the "anomaly" is responsible for the projectiveness of the représentations.

A well-known folklore theorem states that ordinary 2d TFT are classified by (commutative) Frobenius algebras. In a similar way, any modular functor yields a "categorified Frobenius algebra", of which ribbon categories form a large class of examples. In this talk, we'll explain a necessary and sufficient condition for such a structure to extend to a modular functor, formulated in terms of certain generalized skein modules attached to handlebodies. A key observation is that this is, indeed, a condition, not extra structure, so that such an extension is essentially unique whenever it exists.

This construction should be thought of as a far reaching generalization of the construction by Masbaum and Roberts of a modular functor from Kauffman skein modules. As a special case it also recovers, in a purely topological way, the construction of a modular functor from a (not necessarily semisimple) modular category by Lyubachenko, and the uniqueness result is new even in those cases. This is based on joint work with Lukas Woike.