(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
One may say that ‘fibre bundles are fibrations’ by the Milnor slide trick.
In the most general sense, a bundle over an object $B$ in a category $C$ is a morphism $p: E \to B$ in $C$.
In appropriate contexts, a fibre bundle over $B$ with standard fibre $F$ may be defined as a bundle over $B$ such that, given any global element $x: 1 \to B$, the pullback of $E$ along $x$ is isomorphic to $F$. Certainly this definition is appropriate whenever $C$ has a terminal object $1$ which is a separator, 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 $F \to E \to B$ or
even though there is not a single morphism $F \to E$ but instead one for each global element $x$ (and none at all if $B$ has no global elements!).
If $C$ is a site, then a locally trivial fibre bundle over $B$ with typical fibre $F$ is a bundle over $B$ with a cover $(j_\alpha: U_\alpha \to B)_\alpha$ such that, for each index $\alpha$, the pullback $E_\alpha$ of $E$ along $j_\alpha$ is isomorphic in the slice category $C/{U_\alpha}$ to the trivial bundle $U_\alpha \times F$ (a local trivialization).
One can also drop $F$ and define a slightly more general notion of locally trivial bundle over $B$ as a bundle over $B$ with a cover $(j_\alpha: U_\alpha \to B)_\alpha$ such that, for each index $\alpha$, there is a fibre $F_\alpha$ and an isomorphism in $C/{U_\alpha}$ between the pullback $E_\alpha$ and the trivial bundle $U_\alpha \times F_\alpha$. Every locally trivial fibre bundle is obviously a locally trivial bundle; the converse holds if $B$ is connected.
Now suppose that $E$ is a fibre bundle over $B$ with standard fibre $F$, locally trivialised using the cover $(U_\alpha)_\alpha$. Given an index $\alpha$ and an index $\beta$, let $U_{\alpha,\beta}$ be the fibred product (pullback) of $U_\alpha$ and $U_\beta$. Then we have an automorphism $g_{\alpha,\beta}$ of $U_{\alpha,\beta} \times F$ in $C/{U_{\alpha,\beta}}$ as follows: (diagram to come) The $g_{\alpha,\beta}$ are the transition morphisms of the locally trivial fibre bundle $E$.
Often one considers special kinds of bundles, by requiring structure on the standard fibre $F$ and/or conditions on the transition morphisms $g_{\alpha,\beta}$. For example:
If $G$ is a group object in $C$ that acts on $F$, then a $G$-bundle (or bundle with structure group $G$) over $B$ with standard fibre $F$ is a locally trivial fibre bundle over $B$ with standard fibre $F$ together with morphisms $U_{\alpha,\beta} \to G$ that, relative to the action of $G$ on $F$, give the transition maps $g_{\alpha,\beta}$. (The morphism $U_{\alpha,\beta} \to G$ is also written $g_{\alpha,\beta}$, conflating action with application.)
More specifically, a (right or left) principal $G$-bundle over $B$ is a $G$-bundle over $B$ with standard fibre $G$, associated with the action of $G$ on itself by (right or left) multiplication.
If $F$ is an object of a concrete category over $C$, then we can consider locally trivial fibre bundles with standard fibre $F$ such that the transition morphisms are structure-preserving morphisms. If the automorphism group $Aut(F)$ can be internalised in $C$, then this the same as an $Aut(F)$-bundle, but the concept makes sense in any case.
As a fairly specific example, if $F$ is a topological vector space (and $C$ is a category with structure to support this, such as Top or Diff), then a vector bundle over $B$ with standard fibre $F$ is a $GL(F)$-bundle over $B$ with standard fibre $F$, where $GL(F)$ is the general linear group with its defining action on $F$.
Given a right principal $G$-bundle $\pi: P\to X$ and a left $G$-space $F$, all in a sufficiently strong category $C$ (such as Top), one can form the quotient object $P\times_G F = (P\times F)/{\sim}$, where $P \times F$ is a product and $\sim$ is the smallest congruence such that (using generalized elements) $(p g,f)\sim (p,g f)$; there is a canonical projection $P\times_G F\to X$ where the class of $(p,f)$ is mapped to $\pi(p)\in X$, hence making $P\times_G F\to X$ into a fibre bundle with typical fiber $F$, and the transition functions belonging to the action of $G$ on $F$. We say that $P\times_G F\to X$ is the associated bundle to $P\to X$ with fiber $F$.
In higher category theory the notion of fiber bundle generalizes. See
Under the interpretation of modules as generalized vector bundles, locally trivial fiber bundles correspond to locally free modules. See there for more.
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
principal bundle / torsor / associated bundle
Norman Steenrod, section I.7 of The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)
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.) (pdf)
Dale Husemoeller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Basic Bundle Theory and K-Cohomology Invariants,
Lecture Notes in Physics, Springer 2008 (pdf)
Ralph Cohen, The Topology of Fiber Bundles, Stanford University (2017) (pdf, OMN:201707.110706)
With an eye towards application in mathematical physics:
Last revised on October 19, 2020 at 05:49:02. See the history of this page for a list of all contributions to it.