In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles. For example, one can consider a bundle π: E→ B with E and Bsets. It is no longer true that the preimages must all look alike, unlike fiber bundles where the fibers must all be isomorphic and homeomorphic.
Definition
A bundle is a triple where are sets and a map.
is called the total space
is the base space of the bundle
is the projection
This definition of a bundle is quite unrestrictive. For instance, the empty function defines a bundle. Nonetheless it serves well to introduce the basic terminology, and every type of bundle has the basic ingredients of above with restrictions on and usually there is additional structure. For each is the fibre or fiber of the bundle over. A bundle is a subbundle of if and. A cross section is a map such that for each, that is,.
Examples
If and are smooth manifolds and is smooth, surjective and in addition a submersion, then the bundle is a fibered manifold. Here and in the following examples, the smoothness condition may be weakened to continuous or sharpened to analytic, or it could be anything reasonable, like continuously differentiable, in between.
If for each two points and in the base, the corresponding fibers and are homotopy equivalent, then the bundle is a fibration.
If for each two points and in the base, the corresponding fibers and are homeomorphic, and in addition the bundle satisfies certain conditions of local triviality outlined in the pertaining linked articles, then the bundle is a fiber bundle. Usually there is additional structure, e.g. a group structure or a vector space structure, on the fibers besides a topology. Then is required that the homeomorphism is an isomorphism with respect to that structure, and the conditions of local triviality are sharpened accordingly.
If for each two points and in the base, the corresponding fibers and are vector spaces of the same dimension, then the bundle is a vector bundle if the appropriate conditions of local triviality are satisfied. The tangent bundle is an example of a vector bundle.
Bundle objects
More generally, bundles or bundle objects can be defined in any category: in a categoryC, a bundle is simply an epimorphism π: E → B. If the category is not concrete, then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category with pullbacks and a terminal object 1 the points of B can be identified with morphisms p:1→B and the fiber of p is obtained as the pullback of p and π. The category of bundles over B is a subcategory of the slice category of objects over B, while the category of bundles without fixed base object is a subcategory of the comma category which is also the functor categoryC², the category of morphisms in C. The category of smooth vector bundles is a bundle object over the category of smooth manifolds in Cat, the category of small categories. The functortaking each manifold to its tangent bundle is an example of a section of this bundle object.
