A connection on a bundle can be introduced in several very different but equivalent formalisms: as a parallel transport functor, as a rule for covariant derivative, as a distribution (field) of horizontal subspaces (see Ehresmann connection) and via a connection $1$-form which annihilates the distribution of horizontal subspaces. The connection in that sense induces a smooth version of Hurewicz connection. 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.

The following article mostly deals with parallel transport.

Contents

Idea

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

in $X$ (which is a morphism in the path groupoid $P_1(X)$) a morphism $\mathrm{tra}(\gamma )$ between the fibers of $P$ over these points

such that

• 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);

• this assignment is smooth in a suitable sense;

• this assignment is functorial in that for all pairs $x\stackrel{\gamma }{\to }y$, $y\stackrel{\gamma \prime }{\to }z$ of composable paths in $X$ we have

  $$\begin{array}{ccccc}{P}_x& \stackrel{\mathrm{tra}(\gamma )}{\to }& P_y& \stackrel{\mathrm{tra}(\gamma \prime )}{\to }& P_z\\ x& \stackrel{\gamma }{\to }& y& \stackrel{\gamma \prime }{\to }& z\end{array}=\begin{array}{ccc}{P}_x& \stackrel{\mathrm{tra}(\gamma \prime \circ \gamma )}{\to }& P_z\\ x& \stackrel{\gamma \prime \circ \gamma }{\to }& z\end{array}$$\array{ P_x &\stackrel{tra(\gamma)}{\to}& P_y &\stackrel{tra(\gamma')}{\to}& P_z \\ x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z } \;\;\; = \;\;\; \array{ P_x &\stackrel{tra(\gamma' \circ \gamma)}{\to}& P_z \\ x &\stackrel{\gamma'\circ \gamma}{\to}& z }

In other words, a connection on $P$ is a functor

from the path groupoid of $X$ to the Atiyah Lie groupoid of $P$ that is smooth in a suitable sense and splits the Atiyah sequence in that $P_1(X)\stackrel{\mathrm{tra}}{\to }\mathrm{At}″(X)\to P_1(X)$ (see the notation at Atiyah Lie groupoid).

Terminology

The functor $\mathrm{tra}$ is called the parallel transport of the connection. This terminology comes from looking at the orbits of all points in $P$ under $\mathrm{tra}$ (i.e. from looking at the category of elements of $\mathrm{tra}$): 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 $\mathrm{tra}(\gamma )$ as the result of propagating $p$ parallel to these curves in $P$.

flat connections

It may happen that the assignment $\mathrm{tra}:\gamma \mapsto \mathrm{tra}(\gamma )$ only depends on the homotopy class of the path $\gamma$ relative to its endpoints $x,y$. In other words: that $\mathrm{tra}$ 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 flat connection.

more concrete picture

By Lie differentiation the functor $\mathrm{tra}$, i.e. by looking at what it does to very short pieces of paths, one obtains from it a splitting of the Atiyah Lie algebroid sequence, which is a morphism

of vector bundles. Locally on $X$ – meaning: when everything is pulled back to a cover $Y\to X$ of $X$ – this is a $\mathrm{Lie}(G)$-valued 1-form $A\in {\Omega }^1(Y,\mathrm{Lie}(G))$ 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 Ehresmann connection.

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

from the space $\Gamma (E)$ of section of $E$ that satisfies the properties of a covariant derivative.

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

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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

(namely a morphism in the right (infinity,1)-category which may be represented by an anafunctor).

It turns out that similarly suitable morphisms

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.

(…am running out of time… for more see here) …

Definition

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Special cases

connections on the tangent bundle

Connections on tangent bundles play an important role for instance on Riemannian manifolds and pseudo-Riemannian manifolds. 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 symbols.

Generalizations

superconnections

Generalizing the parallel transport definition from ordinary manifolds to supermanifolds yields the notion of superconnection.

Simons-Sullivan 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.

References

The description of connections as smooth functors on the path groupoid has a long history in its restriction to based loops . This is recalled at

• Urs Schreiber, History of Understanding Bundles with Connection using Parallel Transport around Loops.

The idea to generalize this from loops to paths is due to John Baez and appears in

• John Baez, Urs Schreiber, Higher Gauge Theory (arXiv)

Full details, including detailed discussion of the equivalence to other familiar definitions, are in

• Urs Schreiber, Konrad Waldorf, Parallel transport and functors (arXiv)

A proposal for the full proper generalization of the functorial notion of connection to the context of smooth infinity-stacks along the lines described at motivation for sheaves, cohomology and higher stacks is in section 2.1.6 of

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential String- and Fivebrane structures (pdf)

For more links, more history and further pointers see

• Differential Nonabelian Cohomology .