Given any modular category over an algebraically closed field k, we extract a sequence (Mg)g≥0 of -bimodules. We show that the Hochschild chain complex CH(;Mg) of with coefficients in Mg carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus g+1. The ordinary Hochschild complex of corresponds to CH(;M0). This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor 𝔉:-𝖲𝗎𝗋𝖿𝖼→𝖢𝗁k with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in . The functor 𝔉 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.