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 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 (*E*, *p*, *B*) where *E*, *B* are sets and *p*:*E*→*B* is a map.^{[1]}

*E*is called the**total space***B*is the**base space**of the bundle*p*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 *E*, *p*, *B* and usually there is additional structure.

For each *b* ∈ *B*, *p*^{−1}(*b*) is the **fibre** or **fiber** of the bundle over *b*.

A bundle (*E**, *p**, *B**) is a **subbundle** of (*E*, *p*, *B*) if *B** ⊂ *B*, *E** ⊂ *E* and *p** = *p*|_{E*}.

A cross section is a map *s*:*B* → *E* such that *p*(*s*(*b*)) = *b* for each *b* ∈ *B*, that is, *s*(*b*) ∈ *p*^{−1}(*b*).

## Examples

- If
*E*and*B*are smooth manifolds and*p*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 (*C*^{1}), in between. - If for each two points
*b*_{1}and*b*_{2}in the base, the corresponding fibers*p*^{−1}(*b*_{1}) and*p*^{−1}(*b*_{2}) are homotopy equivalent, then the bundle is a fibration. - If for each two points
*b*_{1}and*b*_{2}in the base, the corresponding fibers*p*^{−1}(*b*_{1}) and*p*^{−1}(*b*_{2}) 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. - A principal bundle is a fiber bundle endowed with a right group action with certain properties. One example of a principal bundle is the frame bundle.
- If for each two points
*b*_{1}and*b*_{2}in the base, the corresponding fibers*p*^{−1}(*b*_{1}) and*p*^{−1}(*b*_{2}) 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 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 **C**², 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.

## See also

## Notes

**^**Husemoller 1994 p 11.

## References

- Goldblatt, Robert (2006) [1984].
*Topoi, the Categorial Analysis of Logic*. Dover Publications. ISBN 978-0-486-45026-1. Retrieved 2009-11-02. - Husemoller, Dale (1994) [1966],
*Fibre bundles*, Graduate Texts in Mathematics,**20**, Springer, ISBN 0-387-94087-1 - Vassiliev, Victor (2001) [2001],
*Introduction to Topology*, Student Mathematical Library, Amer Mathematical Society, ISBN 0821821628