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.
Keywords: n-fold vector bundle atlas, linear decomposition
2020 MSC: 53C05 (Primary), 18F15, 55R65 (Secondary)
Theory and Applications of Categories, Vol. 35, 2020, No. 19, pp 665-699.