Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories

Given any modular category C over an algebraically closed field k, we extract a sequence (M_g)_g≥0 of C-bimodules and show that the Hochschild chain complex CH(C;M_g) of C with coefficients in M_g carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus g+1. The ordinary Hochschild complex of C corresponds to CH(C;M₀). This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor F_C:C-Surf^c⟶Ch_k with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in C. The functor F_C satisfies an excision property which is formulated in terms of homotopy coends. In this sense, any modular category gives naturally rise to a modular functor with values in chain complexes. In zeroth homology, it recovers Lyubashenko's mapping class group representations. The chain complexes in our construction are explicitly computable by choosing a marking on the surface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connected and simply connected groupoid of cut systems that appears in the Lego-Teichmüller game by a contractible Kan complex.