Bundle (mathematics)

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 "B" sets. It is no longer true that the preimages formula_1 must all look alike, unlike fiber bundles where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic.
Definition.
A bundle is a triple where are sets and is a map.
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, .
Bundle objects.
More generally, bundles or bundle objects can be defined in any category: in a category C, 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 (C↓"B") of objects over "B", while the category of bundles without fixed base object is a subcategory of the comma category ("C"↓"C") which is also the functor category C², 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 functor taking each manifold to its tangent bundle is an example of a section of this bundle object.