Given any modular category $\mathcal{C}$ over an algebraically closed field k, we extract a sequence ${(M_g)}_{g\ge 0}$ of $\mathcal{C}$-bimodules and show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ 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 $\mathcal{C}$ corresponds to $CH(\mathcal{C};M_0)$.

This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor ${\mathfrak{F}}_{\mathcal{C}}:\mathcal{C}\text{-}{\mathsf{Surf}}^{\mathsf{c}}⟶{\mathsf{Ch}}_k$ with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in $\mathcal{C}$. The functor ${\mathfrak{F}}_{\mathcal{C}}$ 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.

给定任何模块化类别 $\mathcal{C}$在代数闭域k 上，我们提取一个序列${({米}_G)}_{G\ge 0}$ 的 $\mathcal{C}$-bimodules 并显示 Hochschild 链复合体 $CH(\mathcal{C};{米}_G)$ 的 $\mathcal{C}$ 系数在 ${米}_G$ 携带属表面的映射类群的典型同伦相干射影作用 $G+1$. 普通的 Hochschild 复合体$\mathcal{C}$ 对应于 $CH(\mathcal{C};{米}_0)$.

这个结果是作为以下更全面的拓扑结构的一部分获得的：我们构造了一个对称幺半群函子 ${\mathfrak{F}}_{\mathcal{C}}：\mathcal{C}\text{——}{\mathsf{冲浪}}^{\mathsf{C}}⟶{\mathsf{通道}}_{克}$在k上的链复数中的值定义在对称的幺半群表面上，其边界分量用投影对象标记$\mathcal{C}$. 函子${\mathfrak{F}}_{\mathcal{C}}$满足以同伦共尾形式表示的切除性质。从这个意义上说，任何模范畴都会自然地产生一个在链复合体中具有值的模函子。在第零同源中，它恢复了 Lyubashenko 的映射类群表示。

通过选择表面上的标记，即切割系统和某个嵌入图，我们构造中的链复合体是可明确计算的。为了我们的证明，我们用可收缩的 Kan 复合体替换了 Lego-Teichmüller 游戏中出现的切割系统的连通和简单连通群。