The Hochschild complex of a finite tensor category
We express the Hochschild complex of a finite tensor category using a specific projective resolution of the canonical coend of the finite tensor category. This leads to our main result that the Hochschild complex of a not necessarily semisimple modular category carries a canonical homotopy coherent projective action of the mapping class group $\text{SL}(2,\mathbb{Z})$ of the torus. This supports the idea to think of the Hochschild complex of a modular category as a derived conformal block for the torus. In the semisimple case, the (underived) conformal block is known to carry a commutative multiplication. For the derived conformal block (i.e. in the non-semisimple case), we generalize this to an $E_2$-structure. Our results allow us obtain a notion of Hochschild chains for braided crossed monoidal categories in the sense of Turaev. We prove that these admit an action of an operad built from Hurwitz spaces.