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\begin{document}

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\section*{connection on a bundle}

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

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{more_concrete_picture}{More concrete picture}\dotfill \pageref*{more_concrete_picture} \linebreak
\noindent\hyperlink{more_abstract_picture}{More abstract picture}\dotfill \pageref*{more_abstract_picture} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{ExistenceOfConnections}{Existence of connections}\dotfill \pageref*{ExistenceOfConnections} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{connections_on_the_tangent_bundle}{Connections on the tangent bundle}\dotfill \pageref*{connections_on_the_tangent_bundle} \linebreak
\noindent\hyperlink{connections_in_physics}{Connections in physics}\dotfill \pageref*{connections_in_physics} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{superconnections}{Superconnections}\dotfill \pageref*{superconnections} \linebreak
\noindent\hyperlink{simonssullivan_structured_bundles}{Simons-Sullivan structured bundles}\dotfill \pageref*{simonssullivan_structured_bundles} \linebreak
\noindent\hyperlink{connections_on_a_principal_bundle}{Connections on a principal $\infty$-bundle}\dotfill \pageref*{connections_on_a_principal_bundle} \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{connection} on a [[bundle]] $P \to X$ -- a [[principal bundle]] or an [[associated bundle]] like a [[vector bundle]] -- is a rule that identifies [[fiber]]s of the bundle along paths in the base space $X$.

There are several different but equivalent formalizations of this idea:

\begin{itemize}%
\item as a \textbf{[[parallel transport]]} functor,


\item as a rule for a \textbf{[[covariant derivative]]},


\item as a \textbf{distribution (field) of horizontal subspaces} -- an [[Ehresmann connection]] -- and via a \textbf{connection $1$-form} which annihilates the distribution of horizontal subspaces. The connection in that sense induces a smooth version of [[Hurewicz connection]].



\end{itemize}
The usual textbook convention is to say just connection for the distribution of horizontal subspaces, and the objects of the other three approaches one calls more specifically covariant derivative, connection $1$-form and [[parallel transport]].

In the remainder of this Idea-section we discuss a bit more how to understand connections in terms of parallel transport.

Given a smooth [[bundle]] $P \to X$, for instance a $G$-[[principal bundle]] or a [[vector bundle]], a \emph{connection} on $P$ is a prescription to associate with each path

\begin{displaymath}
\gamma : x \to y
\end{displaymath}
in $X$ (which is a [[morphism]] in the [[path groupoid]] $\mathbf{P}_1(X)$) a morphism $tra(\gamma)$ between the fibers of $P$ over these points

\begin{displaymath}
\itexarray{
     P_x &\stackrel{tra(\gamma)}{\to}& P_y
     \\
     x &\stackrel{\gamma}{\to}& y
  }
\end{displaymath}
such that

\begin{itemize}%
\item this assignment respects the structure on the fibers $P_x$ (for instance is $G$-equivariant in the case that $P$ is a $G$-bundle or that is linear in the case that $P$ is a vector bundle);


\item this assignment is smooth in a suitable sense;


\item this assignment is [[functor]]ial in that for all pairs $x \stackrel{\gamma}{\to} y$, $y \stackrel{\gamma'}{\to} z$ of composable paths in $X$ we have

\begin{displaymath}
\itexarray{
     P_x &\stackrel{tra(\gamma)}{\to}& P_y &\stackrel{tra(\gamma')}{\to}& P_z
     \\
     x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z
  }
  \;\;\;
  =
  \;\;\;
  \itexarray{
     P_x &\stackrel{tra(\gamma' \circ \gamma)}{\to}& P_z
     \\
     x &\stackrel{\gamma'\circ \gamma}{\to}& z
  }
\end{displaymath}


\end{itemize}
In other words, a connection on $P$ is a [[functor]]

\begin{displaymath}
tra : \mathbf{P}_1(X) \to At''(P)
\end{displaymath}
from the [[path groupoid]] of $X$ to the [[Atiyah Lie groupoid]] of $P$ that is smooth in a suitable sense and \emph{splits the Atiyah sequence} in that $\mathbf{P}_1(X) \stackrel{tra}{\to} At''(X) \to \mathbf{P}_1(X)$ (see the notation at [[Atiyah Lie groupoid]]).

\textbf{Terminology}

The functor $tra$ is called the \textbf{[[parallel transport]]} of the connection. This terminology comes from looking at the orbits of all points in $P$ under $tra$ (i.e. from looking at the [[category of elements]] of $tra$): these trace out paths in $P$ sitting over paths in $X$ and one thinks of the image of a point $p \in P_x$ under $tra(\gamma)$ as the result of propagating $p$ parallel to these curves in $P$.

\textbf{Flat connections}

It may happen that the assignment $tra : \gamma \mapsto tra(\gamma)$ only depends on the [[homotopy]] class of the path $\gamma$ relative to its endpoints $x, y$. In other words: that $tra$ factors through the functor $P_1(X) \to \Pi_1(X)$ from the [[path groupoid]] to the [[fundamental groupoid]] of $X$. In that case the connection is called a \textbf{[[flat connection]]}.

\hypertarget{more_concrete_picture}{}\subsubsection*{{More concrete picture}}\label{more_concrete_picture}

By [[Lie theory|Lie differentiation]] the functor $tra$, i.e. by looking at what it does to very short pieces of paths, one obtains from it a splitting of the [[Atiyah Lie groupoid|Atiyah Lie algebroid sequence]], which is a morphism

\begin{displaymath}
\nabla : T X \to at(P)
\end{displaymath}
of vector bundles. Locally on $X$ -- meaning: when everything is pulled back to a [[cover]] $Y \to X$ of $X$ -- this is a $Lie(G)$-valued 1-form $A \in \Omega^1(Y, Lie(G))$ with certain special properties.

In particular, since every $G$-[[principal bundle]] canonically trivializes when pulled back to its own total space $P$, a connection in this case gives rise to a 1-form $A \in \Omega^1(P)$ satisfying two conditions. This data is called an \textbf{[[Ehresmann connection]]}.

If instead $P = E$ is a vector bundle, then the two conditions satisfies by $A$ imply that it defines a linear map

\begin{displaymath}
\nabla : \Gamma(E) \to \Omega^1(X) \otimes \Gamma(E)
\end{displaymath}
from the space $\Gamma(E)$ of section of $E$ that satisfies the properties of a \textbf{covariant derivative}.

If again the connection is flat, then this is manifestly the datum of a [[Lie infinity-algebroid representation]] of the [[Lie algebroid|tangent Lie algebroid]] $T X$ of $X$ on $E$: it defines the action [[Lie algebroid]] which is the [[Lie theory|Lie version]] of the [[Lie groupoid]] classified by the parallel transport functor.

\ldots{}

\hypertarget{more_abstract_picture}{}\subsubsection*{{More abstract picture}}\label{more_abstract_picture}

Recall from the discussion at $G$-[[principal bundle]] that the $G$-bundle $P \to X$ is encoded in a a suitable morphism

\begin{displaymath}
X \to \mathbf{B}G
\end{displaymath}
(namely a morphism in the right [[(infinity,1)-category]] which may be represented by an [[anafunctor]]).

It turns out that similarly suitable morphisms

\begin{displaymath}
\mathbf{P}_1(X) \to \mathbf{B}G
\end{displaymath}
encode in one step the $G$-bundle together with its [[parallel transport]] functor.

This is described in great detail in the reference by Schreiber--Waldorf below.

(\ldots{}am running out of time\ldots{} )

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

Let $G$ be a [[Lie group]]. We recall briefly the following discussion of $G$-[[principal bundle]]s. For an in-depth discussion see [[Smooth∞Grpd]].

Write

\begin{displaymath}
\mathbf{B}G : U \mapsto ( Hom_{Diff}(U,G) \stackrel{\to}{\to} *)
\end{displaymath}
for the [[functor]] that sends a [[Cartesian space]] $U$ to the [[delooping]] [[groupoid]] of the group of $G$-valued [[smooth function]]s on $U$: the groupoid with a single object and the [[group]] $Hom_{Diff}(U,G)$ of maps as its set of morphisms.

This is a groupoid-valued [[sheaf]] on the [[site]] [[CartSp]]${}_{smooth}$ and in fact is a [[(2,1)-sheaf]]/[[stack]].

For $X$ a [[paracompact space|paracompact]] [[smooth manifold]], we may also regard it as a [[(2,1)-sheaf]] on [[CartSp]] in an evident way.

\begin{prop}
\label{}\hypertarget{}{}
The groupoid $G Bund(X)$ of $G$-[[principal bundle]]s on $X$ is equivalent to the hom-groupoid

\begin{displaymath}
\mathbf{H}(X,\mathbf{B}G) \simeq G Bund(X)
\end{displaymath}
taken in the [[(2,1)-topos]] of [[(2,1)-sheaves]] on [[CartSp]]${}_{smooth}$.

\end{prop}
A detailed discussion of this is at [[Smooth∞Grpd]] in the section .

Now write $\mathfrak{g}$ for the [[Lie algebra]] of $\mathfrak{g}$. Then consider the functor

\begin{displaymath}
\mathbf{B} G_{conn} 
  : U \mapsto 
  [\mathbf{P}_1(U),\mathbf{B}G]
  =
  \left\{
    A \stackrel{g}{\to} (g^{-1} A g + g^{-1} d g)
    |
    A \in \Omega^1(U,\mathfrak{g})\,, g \in C^\infty(U,G)
  \right\}
\end{displaymath}
that sends a [[Cartesian space]] $U$ to the \textbf{[[groupoid of Lie-algebra valued 1-forms]]} over $U$.

There is an evident morphism of [[(2,1)-sheaves]]

\begin{displaymath}
\mathbf{B}G_{conn} \to \mathbf{B}G
\end{displaymath}
that forgets the 1-forms on each object $U$.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(connection)}

A \textbf{connection} on a smooth $G$-[[principal bundle]] $g : X \to \mathbf{B}G$ is a lift $\nabla$ to $\mathbf{B}G_{conn}$

\begin{displaymath}
\itexarray{
    && \mathbf{B}G_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,.
\end{displaymath}
The \textbf{groupoid of $G$-principal bundles with connection} on $X$ is

\begin{displaymath}
G Bund_\nabla(X) := Hom(X,\mathbf{B}G_{conn})
  \,.
\end{displaymath}
\end{defn}
Explicitly, a morphism $g : X \to \mathbf{B}G$ is a nonabelian [[Cech cohomology]] [[cocycle]] on $X$ with values in $G$:

\begin{enumerate}%
\item a choice of [[good open cover]] $\{U_i \to X\}$ of $X$;


\item a collection of [[smooth function]]s $(g_{i j} \in C^\infty(U_i \cap U_j), G)$



\end{enumerate}
such that on $U_i \cap U_j \cap U_k$ the equation

\begin{itemize}%
\item $g_{i j} g_{j k} = g_{i k}$

\end{itemize}
holds.

A lift $\nabla : X \to \mathbf{B}G_{conn}$ of this is in addition

\begin{enumerate}%
\item a choice of [[Lie-algebra valued 1-form]]s $(A_i \in \Omega^1(U_i, \mathfrak{g}))$

\end{enumerate}
such that on $U_i \cap U_j$ the equation

\begin{itemize}%
\item $A_j = g^{-1} A_i g + g^{-1} d g$

\end{itemize}
holds, where on the right we have the pullback $g^* \theta$ of the [[Maurer-Cartan form]] on $G$ (see there).

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{ExistenceOfConnections}{}\subsubsection*{{Existence of connections}}\label{ExistenceOfConnections}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(existence of connections)}

Every $G$-principal bundle admits a connection. In other words, the forgetful functor

\begin{displaymath}
Hom(X, \bar \mathbf{B}G_{conn}) \to Hom(X,\mathbf{B}G)
\end{displaymath}
is an [[essentially surjective functor]].

\end{defn}
\begin{proof}
Choose a [[partition of unity]] $(\rho_i \in C^\infty(X,\mathbb{R}))$ subordinate to the [[good open cover]] $\{U_i \to X\}$ with respect to which a given cocycle $g : X \to \mathbf{B}G$ is expressed:

\begin{itemize}%
\item $(x \;not\; in\; U_i) \Rightarrow \rho_i(x) = 0$;


\item $\sum_i \rho_i = 1$.



\end{itemize}
Then set

\begin{displaymath}
A_i := \sum_{i_0} \rho_{i_0}|_{U_{i_0}} (g_{i_0 i}|^{-1}_{U_{i_0}}) d_{dR} (g_{i_0 i}|_{U_{i_0}})
  \,.
\end{displaymath}
By slight abuse of notation we shall write this and similar expressions simply as

\begin{displaymath}
A_i := \sum_{i_0} \rho_{i_0}(g_{i_0 i}^{-1} d_{dR} g_{i_0 i})  
  \,.
\end{displaymath}
Using the that $(g_{i j})$ satisfies its cocycle condition, one checks that this satisfies the cocycle condition for the 1-forms:

\begin{displaymath}
\begin{aligned}
     A_j - g_{i j}^{-1} A_i g_{i j}  &=
     \sum_{i_0} \rho_{i_0} ( g_{i_0 j}^{-1} d g_{i_0 j} -  
       ( g_{i_0 i} g_{i j}) ^{-1} (d g_{i_0 i}) g_{i j} )
     \\
     & = 
     \sum_{i_0} \rho_{i_0} ( g_{i j}^{-1} d g_{i j} )
     \\
     & =
     g_{i j}^{-1} d g_{i j}
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\hypertarget{connections_on_the_tangent_bundle}{}\subsubsection*{{Connections on the tangent bundle}}\label{connections_on_the_tangent_bundle}

Connections on [[tangent bundle]]s are also called [[affine connection]]s, or [[Levi-Civita connection]]s.

They play a central role for instance on [[Riemannian manifold]]s and [[pseudo-Riemannian metric|pseudo-Riemannian manifold]]s. From the end of the 19th century and the beginning of the 20th centure originates a language to talk about these in terms of [[Christoffel symbol]]s.

\hypertarget{connections_in_physics}{}\subsubsection*{{Connections in physics}}\label{connections_in_physics}

In [[physics]] connections on bundles model [[gauge field]]s.

\begin{itemize}%
\item The [[electromagnetic field]] is a connection on a [[circle group]]-principal bundle;


\item A [[Yang-Mills field]] more generally is a connection on a [[unitary group]]-principal bundle.


\item The field of [[gravity]] is encoded in a connection on the [[orthogonal group]]-principal bundle to which the [[tangent bundle]] is [[associated bundle|associated]].



\end{itemize}
For more on this see [[higher category theory and physics]].

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

\hypertarget{superconnections}{}\subsubsection*{{Superconnections}}\label{superconnections}

Generalizing the parallel transport definition from ordinary manifolds to [[supermanifold]]s yields the notion of [[superconnection]].

\hypertarget{simonssullivan_structured_bundles}{}\subsubsection*{{Simons-Sullivan structured bundles}}\label{simonssullivan_structured_bundles}

When the notion of connection on a principal bundle is slightly coarsened, i.e. when more connections are regarded as being ismorphic than usual, one arrives at a structure called a [[Simons-Sullivan structured bundle]]. This has the special property that for $G = U$ the [[unitary group]], the corresponding [[Grothendieck group]] of such bundles is a model for [[differential K-theory]].

\hypertarget{connections_on_a_principal_bundle}{}\subsubsection*{{Connections on a principal $\infty$-bundle}}\label{connections_on_a_principal_bundle}

See [[connection on a principal ∞-bundle]].

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

\begin{itemize}%
\item \textbf{connection on a bundle}, [[universal connection]]

\begin{itemize}%
\item [[principal connection]]

\begin{itemize}%
\item [[affine connection]], [[Levi-Civita connection]] [[Cartan connection]]


\item [[Chen connection]]


\item [[covariant derivative]]


\item [[equivariant connection]]


\item [[logarithmic connection]]



\end{itemize}

\item [[Chern connection]]


\item [[moduli space of connections]]


\item [[fiber bundles in physics]]



\end{itemize}

\item [[connection on a 2-bundle]] / [[connection on a gerbe]] / [[connection on a bundle gerbe]]


\item [[connection on a 3-bundle]] / [[connection on a bundle 2-gerbe]]


\item [[connection on an ∞-bundle]]


\item [[parallel transport]], [[higher parallel transport]]


\item [[holonomy]], [[higher holonomy]]

\begin{itemize}%
\item [[holonomy group]]


\item [[special holonomy]]


\item [[nonabelian Stokes theorem]]



\end{itemize}

\item [[fiber bundles in physics]]



\end{itemize}
[[!include higher Atiyah groupoid - table]]

[[!include gauge field - table]]

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

A classical textbook reference is

\begin{itemize}%
\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973)

\end{itemize}
The formulation of connections in terms of their smooth [[parallel transport]] [[functors]] is in

\begin{itemize}%
\item [[Urs Schreiber]], [[Konrad Waldorf]], \emph{Parallel Transport and Functors}, J. Homotopy Relat. Struct. 4, 187-244 (2009) (\href{http://arxiv.org/abs/0705.0452}{arXiv:0705.0452})

\end{itemize}
based on a series of classical observations. I farily comprehensive commented list of related references is here:

\begin{itemize}%
\item \emph{}

\end{itemize}
Basic facts about connections, such as the existence proof in terms of Cech cocycles, are collected in the brief lecture note

\begin{itemize}%
\item [[Theodore Voronov]], \emph{Differential Geometry, \S{}3 Connection on a vector bundle} (\href{http://www.maths.manchester.ac.uk/~tv/Teaching/Differential%20Geometry/2008-2009/lecture3.pdf}{pdf})

\end{itemize}
Discussion of connection form (mostly on trivial bundles) in [[synthetic differential geometry]] includes the following

\begin{itemize}%
\item [[Anders Kock]], \emph{Connections and path connections in groupoids}, 2006 (\href{http://math.au.dk/publs?publid=619}{web})


\item [[Anders Kock]], \emph{Group valued differential forms revisited}, Feb 2007 (\href{http://math.au.dk/publs?publid=636}{web})



\end{itemize}
[[!redirects connections on a bundle]] [[!redirects connections on bundles]]

[[!redirects bundle with connection]] [[!redirects bundle with connections]] [[!redirects bundles with connection]] [[!redirects bundles with connections]]



\end{document}