We can map from set of coverings over $X$ to symmetric group $\mathfrak{S}_n$ via monodromy (if we fix a loop at the basepoint). Also we can consider braid group $Br_n(Y)$, allow strands pass through themselves and map $Br_n \to \mathfrak{S}_n$ such a way. Can we construct a category of "braided coverings" over $X$ with morphisms to category of coverings over $X$ and to category of subgroups of fixed subrgoup of $Br_n(Y)$ ("$Y$-braided monodromy" group of $X$) such that the square will be commutative?

(edited at 16:48 UTC, June 18)