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 $$\pi^{-1}(x)$$ 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.
 * $E$ 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 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.