We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g \geq 1$ with $n \geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $\mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.

中文翻译：

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从 Hopf monoids 和带状图映射类组动作

我们表明，在对称幺半群类别 $\mathcal{C}$ 中的任何关键 Hopf 幺半群 $H$ 都会引起对具有 $n \geq 1$ 边界分量的类 $g \geq 1$ 的定向表面的类组进行映射的动作. 这些映射类群动作由群同态给出到$H$ 上某些Yetter-Drinfeld 模块的自同构群中。它们与嵌入的带状图中的边缘幻灯片相关联，这些带状图中概括了和弦图中的和弦幻灯片。我们根据生成 Dehn 扭曲和定义关系对这些映射类组动作进行了具体描述。对于 $\mathcal{C}$ 是有限完备和共完备的情况，我们还通过在 Yetter-Drinfeld 模块结构下施加不变性和协方差来获得映射封闭曲面类群的动作。