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\section*{fiber bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{bundle}{$G$-bundle}\dotfill \pageref*{bundle} \linebreak
\noindent\hyperlink{principal_bundles}{$G$-principal bundles}\dotfill \pageref*{principal_bundles} \linebreak
\noindent\hyperlink{structurepreserving_bundles}{Structure-preserving bundles}\dotfill \pageref*{structurepreserving_bundles} \linebreak
\noindent\hyperlink{vector_bundles}{Vector bundles}\dotfill \pageref*{vector_bundles} \linebreak
\noindent\hyperlink{associated_bundles}{Associated bundles}\dotfill \pageref*{associated_bundles} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{in_commutative_algebra}{In commutative algebra}\dotfill \pageref*{in_commutative_algebra} \linebreak
\noindent\hyperlink{in_noncommutative_geometry}{In noncommutative geometry}\dotfill \pageref*{in_noncommutative_geometry} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{fibre bundle} or \textbf{fiber bundle} is a [[bundle]] in which every [[fibre]] is [[isomorphism|isomorphic]], in some coherent way, to a \textbf{standard fibre} (sometimes also called \textbf{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]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

In the most general sense, a \textbf{bundle} over an object $B$ in a [[category]] $C$ is a [[morphism]] $p: E \to B$ in $C$.

In appropriate contexts, a \textbf{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

\begin{displaymath}
\array {
   F & \rightarrow & E \\
     &             & \downarrow \\
     &             & B
}
\end{displaymath}
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 \textbf{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 \emph{[[local trivialization]]}).

One can also drop $F$ and define a slightly more general notion of \textbf{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 object|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 \textbf{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:

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\hypertarget{bundle}{}\subsubsection*{{$G$-bundle}}\label{bundle}

If $G$ is a [[group object]] in $C$ that [[action|acts]] on $F$, then a \textbf{$G$-bundle} (or bundle with \textbf{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.)

\hypertarget{principal_bundles}{}\subsubsection*{{$G$-principal bundles}}\label{principal_bundles}

More specifically, a (right or left) \textbf{[[principal bundle|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.

\hypertarget{structurepreserving_bundles}{}\subsubsection*{{Structure-preserving bundles}}\label{structurepreserving_bundles}

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.

\hypertarget{vector_bundles}{}\subsubsection*{{Vector bundles}}\label{vector_bundles}

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 \textbf{[[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 $F$.

\hypertarget{associated_bundles}{}\subsubsection*{{Associated bundles}}\label{associated_bundles}

Given a right [[principal bundle|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 element]]s) $(p g,f)\sim (p,g f)$; 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 \textbf{[[associated bundle]]} to $P\to X$ with fiber $F$.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\hypertarget{in_higher_category_theory}{}\subsubsection*{{In higher category theory}}\label{in_higher_category_theory}

In [[higher category theory]] the notion of fiber bundle generalizes. See

\begin{itemize}%
\item [[associated ∞-bundle]].

\end{itemize}
\hypertarget{in_commutative_algebra}{}\subsubsection*{{In commutative algebra}}\label{in_commutative_algebra}

Under the interpretation of \href{http://ncatlab.org/nlab/show/module#RelationToVectorBundlesInIntroduction}{modules as generalized vector bundles}, locally trivial fiber bundles correspond to \emph{[[locally free modules]]}. See there for more.

\hypertarget{in_noncommutative_geometry}{}\subsubsection*{{In noncommutative geometry}}\label{in_noncommutative_geometry}

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.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[bundle]]


\item \textbf{fiber bundle} / [[fiber ∞-bundle]]

\begin{itemize}%
\item [[associated bundle]]


\item [[vector bundle]]


\item [[sphere bundle]]


\item [[equivariant bundle]]


\item [[fiber bundles in physics]]


\item [[Leray-Hirsch theorem]]



\end{itemize}

\item [[principal bundle]] / [[torsor]] / \textbf{associated bundle}


\item [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]


\item [[principal 3-bundle]] / [[bundle 2-gerbe]]


\item [[principal ∞-bundle]] / [[associated ∞-bundle]]


\item [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Norman Steenrod]], \emph{The topology of fibre bundles}, Princeton Mathematical Series \textbf{14}, 1951. viii+224 pp. \href{http://www.ams.org/mathscinet-getitem?mr=39258}{MR39258}; reprinted 1994


\item [[Dale Husemöller]], \emph{Fibre bundles}, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. \textbf{20}, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) (\href{http://www.maths.ed.ac.uk/~aar/papers/husemoller}{pdf})


\item [[M M Postnikov]], \emph{  .  IV.  } (Lectures in geometry. Semester IV. Differential geometry.) Nauka, Moscow, 1988. 496 pp. \href{http://www.ams.org/mathscinet-getitem?mr=985587}{MR90h:53002}


\item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf})



\end{itemize}
[[!redirects fibre bundle]] [[!redirects standard fiber]] [[!redirects standard fibre]] [[!redirects typical fiber]] [[!redirects typical fibre]] [[!redirects locally trivial fiber bundle]] [[!redirects locally trivial fibre bundle]] [[!redirects locally trivial bundle]] [[!redirects transition morphism]] [[!redirects transition map]] [[!redirects transition function]] [[!redirects G-bundle]] [[!redirects bundle with structure group]] [[!redirects fiber bundles]] [[!redirects fibre bundles]] [[!redirects standard fibers]] [[!redirects standard fibres]] [[!redirects typical fibers]] [[!redirects typical fibres]] [[!redirects locally trivial fiber bundles]] [[!redirects locally trivial fibre bundles]] [[!redirects locally trivial bundles]] [[!redirects transition morphisms]] [[!redirects transition maps]] [[!redirects transition functions]] [[!redirects G-bundles]] [[!redirects bundles with structure group]] [[!redirects bundles with structure groups]]

[[!redirects locally trivial bundle]] [[!redirects locally trivial bundles]] [[!redirects locally trivial fiber bundle]] [[!redirects locally trivial fiber bundles]]



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