Multiple vector bundles: cores, splittings and decompositions

This paper introduces ∞- and n-fold vector bundles as special functors from the ∞- and n-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of n-fold vector bundles and we prove that any n-fold vector bundle admits a non-canonical isomorphism to a decomposed n-fold vector bundle. A colimit argument then shows that ∞-fold vector bundles admit as well non-canonical decompositions. For the convenience of the reader, the case of triple vector bundles is discussed in detail.