Liftable braids and the coloured braid groupoid

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.