A geometrisation of N-manifolds

Abstract

This paper proposes a geometrisation of $N$-manifolds of degree n as n-fold vector bundles equipped with a (signed) $S_n$-symmetry. More precisely, it proves an equivalence between the categories of $\left[n\right]$-manifolds and the category of (signed) symmetric n-fold vector bundles, by finding that symmetric n-fold vector bundle cocycles and $\left[n\right]$-manifold cocycles are identical.

This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an $S_2$-action.