# 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: EB with E and B sets. It is no longer true that the preimages π − 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:EB 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 bB, 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:BE such that p(s(b)) = b for each bB, that is, s(b) ∈ p−1(b).

## 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: EB. 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 (CB) of objects over B, while the category of bundles without fixed base object is a subcategory of the comma category (CC) 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.