Multiplicity free covering of a graded manifold

We define and study a multiplicity-free covering of a graded manifold. We compute its deck transformation group, which is isomorphic to the permutation group $S_n$. We show that it is not possible to construct a covering of a graded manifold in the category of $n$-fold vector bundles. As an application of our research, we give a new conceptual proof of the equivalence of the categories of graded manifolds and symmetric $n$-fold vector bundles.