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

Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ with coefficients in $M_g$ carries canonical homotopy coherent projective action mapping class group surface genus $g+1$. The ordinary corresponds to $CH(\mathcal{C};M_0)$. This result is obtained as part following more comprehensive topological structure: construct symmetric monoidal functor $\mathfrak{F}_{\mathcal{C}}:\mathcal{C}\text{-}\mathsf{Surf}^{\mathsf{c}}\to\mathsf{Ch}_k$ values complexes $k$ defined on surfaces whose boundary components are labeled objects $\mathcal{C}$. $\mathfrak{F}_{\mathcal{C}}$ satisfies excision property which formulated terms coends. In this sense, gives naturally rise complexes. zeroth homology, it recovers Lyubashenko's representations. our construction explicitly computable by choosing marking surface, i.e. cut system and certain embedded graph. For proof, replace connected simply groupoid systems appears Lego-Teichm\"uller game contractible Kan complex.