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\begin{document}

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\section*{associated bundle}

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

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

[[!include bundles - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak
\noindent\hyperlink{InGeometricHomotopyTheory}{In geometric homotopy theory}\dotfill \pageref*{InGeometricHomotopyTheory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional}

Given a right [[principal bundle|principal]] $G$-bundle $\pi: P\to X$ and a left $G$-[[action]] on some $F$, 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{InGeometricHomotopyTheory}{}\subsubsection*{{In geometric homotopy theory}}\label{InGeometricHomotopyTheory}

In the context of [[higher topos theory]] there is an elegant and powerful definition and construction of associated bundles. We discuss here some basics and how this recovers the traditional definition. For more see at \emph{[[associated infinity-bundle]]} and at \emph{[[geometry of physics -- representations and associated bundles]]}.

At \emph{[[geometry of physics -- principal bundles]]} in the section \emph{\href{geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids}{Smooth principal bundles via smooth groupoids}} is discussed how smooth [[principal bundles]] for a [[Lie group]] $G$ over a [[smooth manifold]] $X$ are equivalently the [[homotopy fibers]] of morphisms of [[smooth groupoids]] ([[smooth stacks]]) of the form

\begin{displaymath}
X \stackrel{}{\longrightarrow} \mathbf{B}G
  \,.
\end{displaymath}
Now given an [[action]] $\rho$ of $G$ on some [[smooth manifold]] $V$, and regardiing this action via its [[action groupoid]] projection $p_\rho \colon V//G \to \mathbf{B}G$ as discussed \hyperlink{ActionsOf1Groups}{above}, then we may consider these two morphisms into $\mathbf{B}G$ jointly

\begin{displaymath}
\itexarray{
    && V//G
    \\
    && \downarrow^{\mathrlap{p_\rho}}
    \\
    X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
  }
\end{displaymath}
and so it is natural to construct their [[homotopy fiber product]].

We now discuss that this is equivalently the [[associated bundle]] which is associated to the principal bundle $P \to X$ via the action $\rho$.

\begin{prop}
\label{}\hypertarget{}{}
For $G$ a [[smooth group]] (e.g. a [[Lie group]]), $X$ a [[smooth manifold]], $P \to X$ a smooth $G$-[[principal bundle]] over $X$ and $\rho$ a smooth [[action]] of $G$ on some [[smooth manifold]] $V$, then the [[associated bundle|associated]] $V$-[[fiber bundle]] $P \times_G V\to X$ is equivalently (regarded as a [[smooth groupoid]]) the [[homotopy pullback]] of the [[action groupoid]]-projection $p_\rho \colon V//G \to \mathbf{B}G$ along a morphism $g \colon X\to\mathbf{B}G$ which [[modulating morphism|modulates]] $P$

\begin{displaymath}
\itexarray{
    P\times_G V &\longrightarrow& V//G
    \\
    \downarrow && \downarrow^{\mathrlap{p_\rho}}
    \\
    X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the discussion at \emph{[[geometry of physics -- principal bundles]]} in the section \emph{\href{geometry%20of%20physics%20--%20principal%20bundles#PrincipalBundlesViaSmoothGroupoids}{Smooth principal bundles via smooth groupoids}}, the morphism $g$ of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an [[open cover]] $\{U_i \to X\}$ over wich $P$ trivialize and choosing a trivialization, by the [[zig-zag]]

\begin{displaymath}
\itexarray{
    C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet
    \\
    \downarrow^{\mathrlap{\simeq_{lwe}}}
    \\
    X
  }
\end{displaymath}
where the top morphism is the [[Cech cohomology|Cech cocycle]] of the given local trivialization regarded as a morphism out of the [[Cech groupoid]] of the given cover.

Moreover, by \href{geometry+of+physics+--+representations+and+associated+bundles#MapFromActionGroupoidOnSetBackToBG}{this proposition} the morphism $(p_\rho)_\bullet$ is a global fibration of pre-smooth groupoids, hence, by the discussion at \emph{[[geometry of physics -- smooth homotopy types]]}, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of $(p_\rho)_\bullet$ along this $g_\bullet$

\begin{displaymath}
\itexarray{
    C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet &\longrightarrow& (V//G)_\bullet
    \\
    \downarrow && \downarrow^{\mathrlap{(p_\rho)_\bullet}}
    \\
    C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}&
    (\mathbf{B}G)_\bullet
    \\
    \downarrow^{\mathrlap{\simeq_{lwe}}}
    \\
    X
  }
  \,.
\end{displaymath}
This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism $(x,i)\to (x,j)$ in the Cech groupoid as well as a morphism $s \stackrel{g}{\to} \rho(s)(g)$ in the action groupoid, such that the group label $g$ of the latter equals the cocycle $g_{i j}(x)$ of the cocycle on the former. Schematically:

\begin{displaymath}
C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet
  =
  \left\{
     ((x,i),s) \stackrel{g_{i j}(x)}{\longrightarrow} ((x,j),\rho(s)(g))
  \right\}
  \,.
\end{displaymath}
This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is [[equivalence of groupoids]] equivalent to the [[quotient]] of its space of objects by the [[equivalence relation]] induced by its morphisms:

\begin{displaymath}
\cdots \simeq
  \left(
    \underset{i}{\coprod} U_i \times V
  \right)/_\sim
  \,.
\end{displaymath}
This is a traditional description of the associated bundle in question.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

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


\item [[tractor bundle]]



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

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

\end{itemize}
[[!redirects associated bundles]]

\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.)



\end{itemize}
[[!redirects associated bundles]]

[[!redirects associated fiber bundle]] [[!redirects associated fiber bundles]]



\end{document}