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 so-called derived modular functors that provide a

consistent family of higher algebraic invariants for a modular tensor

category that are constructed in a way reminiscent of factorization

homology.

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. We explain that the latter construction is the

shadow of a derived modular functor featuring homotopy coherent

mapping class group actions on chain complex valued conformal blocks

and a version of factorization and self-sewing via homotopy coends. On

the torus, we find a derived version of the Verlinde algebra, an

algebra over the little disk operad. The talk is based on joint work

with Christoph Schweigert (Hamburg).

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