Multiplicity free covering of a graded manifold

TL;DR

This paper introduces a multiplicity-free covering for graded manifolds, computes its deck transformation group, and demonstrates the limitations of constructing such coverings within n-fold vector bundles, providing a new proof of their categorical equivalence.

Contribution

It defines a novel multiplicity-free covering for graded manifolds and proves the categorical equivalence with symmetric n-fold vector bundles using this framework.

Findings

Deck transformation group is isomorphic to S_n.

Cannot construct such coverings within n-fold vector bundles.

Provides a new proof of categorical equivalence.

Abstract

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.