Bundle (mathematics)
Generalization of a fiber bundle
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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).
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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 (C1), in between.
- If for each two points b1 and b2 in the base, the corresponding fibers p−1(b1) and p−1(b2) are homotopy equivalent, then the bundle is a fibration.
- If for each two points b1 and b2 in the base, the corresponding fibers p−1(b1) and p−1(b2) 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 b1 and b2 in the base, the corresponding fibers p−1(b1) and p−1(b2) 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.