标题： 可提升的辫子和着色辫子范畴 显示英文标题

标题： Liftable braids and the coloured braid groupoid

摘要： 当$\pi:\widetilde{\Sigma}\rightarrow D^2$是在$n$个标记点上分支的圆盘覆盖时，辫群$B_n$通过固定标记点集的同胚作用在圆盘上。 一个辫子$\beta$ \textit{提升} 如果存在一个同胚$\widetilde{\beta}\in \textit{Mod}(\widetilde{\Sigma})$使得$\beta\circ \pi=\pi\circ \widetilde{\beta}$。 对于任意覆盖，将$\beta$映射到$\widetilde{\beta}$的\textit{提升同态}仅在辫群的一个真子群上定义。 本文将提升同态扩展为从着色辫群胚到映射类群胚的映射，适用于圆盘的所有简单覆盖。 我们表征了每个着色辫的提升，在可提升辫群上恢复了经典的提升同态。 显示英文摘要

摘要： When $\pi:\widetilde{\Sigma}\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $\beta$ \textit{lifts} if there is a homeomorphism $\widetilde{\beta}\in \textit{Mod}(\widetilde{\Sigma})$ such that $\beta\circ \pi=\pi\circ \widetilde{\beta}$. For arbitrary covers, the \textit{lifting homomorphism} taking $\beta$ to $\widetilde{\beta}$ is only defined on a proper subgroup of the braid group. This paper extends the lifting homomorphism to a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc. We characterise the lift of every coloured braid, recovering the classical lifting homomorphism on the liftable braid group.