The Hochschild complex of a finite tensor category

Abstract

Modular functors, ie consistent systems of projective representations of mapping class groups of surfaces, were constructed for nonsemisimple modular categories decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the nonsemisimple case. We elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}\left(2,\mathbb{Z}\right)$ of the torus on the Hochschild complex of a modular category. This is a further step towards understanding the Hochschild complex of a modular category as a differential graded conformal block for the torus. Moreover, we describe a differential graded version of the Verlinde algebra.