doublebundle

Doublebundle is a term sometimes used to describe a bundle of bundles: a fiber bundle E → B whose fiber itself carries a bundle structure over another base. In other words, it is a two-stage fibration with E projecting to B, and the typical fiber F projecting further to a base B'. Formally, one has a bundle p: E → B with fiber F, together with a projection q: F → B', so that the total space E can be viewed as a bundle over B whose fibers are themselves bundles over B'.

The notion emphasizes two levels of geometric structure. If the fiber F → B' is a fiber bundle and E is equipped with a compatible projection p: E → B, then E can often be regarded as a bundle over B' in a way that reflects both bundle structures. When the fiber F is a vector bundle and E inherits a compatible vector bundle structure over B, the construction is frequently called a double vector bundle, a standard object in differential geometry.

Examples are common in product-type constructions. For instance, take E = B × F where F → B' is a fiber bundle. Then E → B is a trivial bundle with fiber F, while F → B' is itself a bundle. This yields a straightforward two-level bundle that illustrates the concept of a doublebundle in a simple setting.

Applications of the idea include hierarchical or staged bundle constructions in geometry and topology, bundle-valued field theories in physics, and the formation of more complex objects such as jet bundles or certain categorical fibrations. See also fiber bundle, bundle of bundles, double vector bundle, and jet bundle for related concepts.