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

TL;DR

This paper extends the geometrization of supergeometry to $Z^r$-graded manifolds of type $ riangle$, establishing an equivalence with a subcategory of $n$-fold vector bundles, generalizing known degree $ extless=2$ cases.

Contribution

It generalizes the geometrization process from degree $ extless=2$ graded manifolds to $Z^r$-graded manifolds of type $ riangle$, linking them to $n$-fold vector bundles.

Findings

Established an equivalence between certain graded manifolds and $n$-fold vector bundles.

Extended the geometrization process to higher $Z^r$-gradings.

Unified the description of complex geometric structures using this new framework.

Abstract

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 $\Delta$.