Graded manifolds of type $\Delta$ and $n$-fold vector bundles

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and graded manifolds of degree $\leq 2$. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of graded manifolds of degree $\leq 2$ and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of $\mathbb Z^r$-graded manifolds of type $\Delta$, where $\Delta$ is a certain weight system and $r$ is the rank of $\Delta$. We establish an equivalence between a subcategory of the category of $n$-fold vector bundles and the category of graded manifolds of type $\Delta$.