A fibre bundle or fiber bundle is a bundle in which every fibre is isomorphic, in some coherent way, to a standard fibre (sometimes also called typical fiber). Though it is pre-dated by many examples and methods, systematic usage of locally trivial fibre bundles with structure groups in mainstream mathematics started with a famous book of Steenrod.
In an appropriate contexts, a fibre bundle over with standard fibre may be defined as a bundle over such that, given any global element , the pullback of along is isomorphic to . Certainly this definition is appropriate whenever has a terminal object which is a generator, as in a well-pointed category; even then, however, one often wants the more restrictive notion below.
One often writes a typical fibre bundle in shorthand as or
even though there is not a single morphism but instead one for each global element (and none at all if has no global elements!).
If is a site, then a locally trivial fibre bundle over with standard fibre is a bundle over with a cover such that, for each index , the pullback of along is isomorphic in the slice category to the trivial bundle .
One can also drop and define a slightly more general notion of locally trivial bundle over as a bundle over with a cover such that, for each index , there is a fibre and an isomorphism in between the pullback and the trivial bundle . Every locally trivial fibre bundle is obviously a locally trivial bundle; the converse holds if is connected.
Now suppose that is a fibre bundle over with standard fibre , locally trivialised using the cover . Given an index and an index , let be the fibred product (pullback) of and . Then we have an automorphism of in as follows: (diagram to come) The are the transition morphisms of the locally trivial fibre bundle .
Often one considers special kinds of bundles, by requiring structure on the standard fibre and/or conditions on the transition morphisms . For example:
If is a group object in that acts on , then a -bundle (or bundle with structure group ) over with standard fibre is a locally trivial fibre bundle over with standard fibre together with morphisms that, relative to the action of on , give the transition maps . (The morphism is also written , conflating action with application.)
More specifically, a (right or left) principal -bundle over is a -bundle over with standard fibre , associated with the action of on itself by (right or left) multiplication.
If is an object of a concrete category over , then we can consider locally trivial fibre bundles with standard fibre such that the transition morphisms are structure-preserving morphisms. If the automorphism group can be internalised in , then this the same as an -bundle, but the concept makes sense in any case.
As a fairly specific example, if is a topological vector space (and is a category with structure to support this, such as Top or Diff), then a vector bundle over with standard fibre is a -bundle over with standard fibre , where is the general linear group with its defining action on .
Given a right principal -bundle and a left -space , all in a sufficiently strong category (such as Top), one can form the quotient object , where is a product and is the smallest congruence such that (using generalized elements) ; there is a canonical projection where the class of is mapped to , hence making into a fibre bundle with typical fiber , and the transition functions belonging to the action of on . We say that is the associated bundle to with fiber .
In higher category theory the notion of fiber bundle generalizes. See
In noncommutative geometry both principal and associated bundles have analogues. The principal bundles over noncommutative spaces typically have structure group replaced by a Hopf algebra; the most well-known class whose base is described by a single algebra are Hopf–Galois extensions; the global sections of the associated bundle are formed using cotensor product. Transition functions can be to some extent emulated using noncommutative localizations, which yield nonaffine generalizations of Hopf–Galois extensions. Another generalization is when Hopf–Galois extensions in the sense of comodule algebras are replaced by entwining structures with analogous Galois condition.
fiber bundle / fiber ∞-bundle
Dale Husemöller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.)