Speaker: Lukas Woike (University of Hamburg)
Title: Derived modular functors
I will present an approach to a class of non-semisimple representation categories, more specifically non-semisimple modular tensor categories, via homotopy theory and low-dimensional topology. This will lead to a so-called derived modular functor, i.e. a consistent system of homotopy coherent mapping class group representations.
It is already well-known that for a semisimple modular tensor category, the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubashenko the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. I will explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complexes which glue via homotopy coends. On the torus, we find a derived version of the Verlinde algebra, an algebra over the little disk operad. The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).