A geometrisation of [formula omitted]-manifolds.

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 [ n ] -manifolds and the category of (signed) 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. [ABSTRACT FROM AUTHOR]