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 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.
A bundle is a triple (E, p, B) where E, B are sets and p:E→B 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).
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.
- Husemoller 1994 p 11.
- Goldblatt, Robert (2006) . Topoi, the Categorial Analysis of Logic. Dover Publications. ISBN 978-0-486-45026-1. Retrieved 2009-11-02.
- Husemoller, Dale (1994) , Fibre bundels, Graduate Texts in Mathematics 20, Springer, ISBN 0-387-94087-1
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