  This paper proposes a geometrisation of $\mathbb N$-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
$[n]$-manifolds and the category of symmetric $n$-fold vector bundles, by
finding that symmetric $n$-fold vector bundle cocycles and $[n]$-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.
