A double vector bundle (DVB) can be defined as a "vector-bundle object in the category of vector bundles". Equivalently, a DVB is a commutative diagram
D -> Bwhere every side is a vector bundle, together with compatibility conditions between the two structures of vector bundle that have D as total space. D can be dualized in two different ways. These two dualization operations do not commute, and generate a group which is isomorphic to the symmetric group S_3. I will explain the above paragraph and then introduce what happens when we replace ``double'' with ``n-fold''. This is joint work with Kirill Mackenzie.
A -> M