For a group (internal to some category, traditionally that of topological spaces) and some other object, a -principal bundle over – also called a -torsor over – is a bundle equipped with a -action on over , such that
the action is principal meaning that
and / or / equivalently (depending on technical details, see below)
and usually it is required that
A central property of -principal bundles over is that they are a geometric model of the degree-1 nonabelian cohomology of with coefficients in . More precisely (subject to some technical details discussed below) there is a natural isomorphism
We discuss first the definition of principal bundles
This is historically and traditionally the default setup. But the theory exists in and is usefully regarded from a more abstract perspective, which, most naturally, is that of a (2,1)-topos. This we introduce and discuss in detail in
we discuss how the traditional setup and many other contexts are recovered from and illuminated by that abstract perspective.
This is the original and oldest branch of the theory. There is a modern established default of the definition, but many slight but crucial variants exists in the literature and are relevant in applications. We start with the modern default notion and then look into its variants.
Let be a topological group.
A -principal bundle over a topological space is a topological space equipped with
such that this is locally trivial in the sense that
A central property of the above definition of principal bundle is
Historically this quotient property of a free continuous action was sometimes taken as the very definition of “principal bundle” without requiring local triviality, e. g. in (Cartan, 1949-1950), where this perspective is attributed to Henri Cartan. A standard modern textbook following this tradition is (Husemöller).
Therefore in order to avoid ambiguous terminology in the following, we will now follow (Palais) and refer to this alternative definition of principal bundle as that of Cartan principal bundle.
Let be a locally compact topological group and a completely regular topological space equipped with a continuous function action . If acts freely on , (no element except the neutral element has any fixed points in under the action) then the map
to the topological quotient is called a -principal bundle in the wide sense. If furthermore the division map
It is no surprise that there is a good theory of principal bundles internal to every topos. However, it turns out that the most “natural home” of the theory is the higher category theoretic context of a (2,1)-topos . This we discuss now, and then relate it to the traditional notion and to various other generalizations.
Notably the existence of universal principal bundles finds its fundamental “explanation” here, where they are seen to be but a presentation of the construction of the homotopy fiber functor, which establishes the equivalence of groupoids
In this context, all of the non-natural aspects of the traditonal theory of principal bundles disappear, for instance
every -principal bundle is locally trivial in a -topos ;
accordingly there is no mismatch between the various definitions anymore as in the context of topological spaces: the condition of principality becomes equivalent to the quotient space condition.
Conversely, the traditional theory nicely naturally embeds into a (2,1)-topos – for instance that of (2,1)-sheaves over the site Top (or rather some small dense subsite thereof) – and the higher topos theory helps to study it there.
The failure of various definitions to match in the traditonal context becomes the fact that the colimits involved get “corrected” to homotopy colimits after embedding into a (2,1)-topos. For instance if an -action on some object is not suitably free, then the -topos theory still produces a healthy principal bundle by replacing the base space by a base groupoid/stack. In fact, this way every action becomes principal over its homotopy quotient. Notably the trivial -action on the terminal object becomes principal over the action groupoid and the resulting -principal bundle is nothing but the universal one.
The following is old material collected from elsewhere that is going to be rearranged….
This indicates the more fundamental way to define -principal bundles in the first place:
Recall (from fiber sequence) that for every group there is the the one-object groupoid . Under the Yoneda embedding this represents a prestack. Write for the corresponding stack obtained by stackification. This is our
This perspective in turn is by general abstract nonsense equivalent to the following useful description:
Let be the suitable (∞,1)-topos internal to which one looks at -principal bundles. For instance for topological bundles this would be Top. For smooth bundles it would be the (∞,1)-category of (∞,1)-sheaves on Diff, etc.
Then every element in is given by a morphism in , which may be thought of as an anafunctor to from the (categorially) discrete category ; the -principal bundle from the beginning of the above definition is just the homotopy pullback of the point along this map, i.e. the homotopy fiber of :
This diagram, incidentally, directly tells us about another important property of -principal bundles: they all canonically trivialize when pulled back to their own total space .
This is what the homotopy commutativity of the above homotopy pullback diagram says: the cocycle pulled back to the bundle that it classifies becomes , which is homotopic to the trivial cocycle (the one that factors through the point) on .
The homotopy pullback here is conveniently and traditionally computed as an ordinary pullback of a fibrant replacement of the pullback diagram. The canonical such fibrant replacement is obtained by replacing by , with an object weakly equivalent to the point, called the -universal principal bundle.
With that the above homotopy pullback is computed as the ordinary pullback
where all squares formed by the lowest horizontal morphisms are homotopy pullback squares, by construction, and where the remaining horizontal morphisms in the top row are induced by the universal property of the homotopy pullback and the morphisms downstairs.
The claim is that
the top row encodes the action of on in that the action is the morphism indicated in
Here the second statement in particular encodes the familiar way to formulate principality of the action , in that it says that
is an isomorphism.
We now unwrap the first statement in gory detail to make clear that this abstract nonsense does reproduce the familiar definition of the action of on .
We now rederive the action of on given just the classifying map by spelling out the details implied by the above abstract description.
Whatever the precise context is (topological, smooth, etc.) we may assume that we are at least in a category of fibrant objects. Then the classifying morphism is represented by an anafunctor, namely a cocycle
The Čech nerve has
morphisms = .
The functor sends
for as described in detail at Čech cohomology.
With the fibrant replacement of the point, which we shall find it helpful to think of as given by
So we read off that is the groupoid with
With determined as an ordinary pullback of a replacement it is convenient for the following to realize it in turn as the pullback-up-to-2-cell in
A moment reflection shows that the component of the natural transformation here is
At the same time recall from the discussion at delooping that the component of the transformation in
Taken together this shows that the universal morphism induced from the commutativity of
and from the homotopy pullback property of
is simply given by the composition of these two component maps
But this is manifestly the right (being both: from the right and correct :-) action of on .
We discuss here aspects of formulating a theory of principal bundles in contexts different from those already discussed above.
For a -principal bundle, its Atiyah Lie groupoid is
with the evident composition operation.
The principal bundle is recovered from its Atiyah Lie groupoid, up to isomorphism, as the source fiber over any point.
This is a classical statement due to Ehresmann … . See for instance (Androulidakis).
Generally, if we accept that we have a large supply of continuous maps between topological spaces, we obtain a -principal bundle on a space for each continuous map to the classifying space of , by pullback of the universal bundle along .
Let be a completely regular topological space and let be a Lie group equipped with a free action on . Then the quotient map is a -principal bundle – in that it is locally trivial – precisely if the division map
is a continuous function.
This is (Palais, theorem 4.1).
Let be the product of infinitly many circles, and let is the product of their order 2 subgroups. This cannot have local section because is locally connected and is not. Therefore is not even locally homeomorphic to .
In fact, the history of the development of the theory of principal bundles and gauge theory is closely related. In the early 1930s Dirac and Hopf independently introduced -principal bundles: Dirac, somewhat implicitly, in his study of the electromagnetic field as a background for quantum mechanics, Hopf in terms of the fibration named after him. However, from there it took apparently many years for the first publication to appear that explicitly states that these two considerations are aspects of the same phenomenon.
gauge field: models and components
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
An original reference on the notion of a principal bundle as a quotient map by a free continuous action of a topological group is
some of which is recollected in (Palais).
A standard textbook following this original perspective is
For principal bundles in the smooth context see most textbooks on differential geometry, for instance
also around section 3.1 of
Lecture notes on the topic include
Discussion of Atiyah Lie groupoids associated to principal bundles and the reconstruction of principal bundles from their Atiyah Lie groupoids is due to
Further discussion along these lines is for instance in
Detailed discussion of topological quotients of groups as principal -bundles is in
Explicit examples and counter examples of coset principal bundles are discussed in
Relations between classes of continuous and of smooth principal bundles are discussed in
Extensions of principal bundles are discussed for instance in
Kirill Mackenzie, On extensions of principal bundles, Annals of Global Analysis and Geometry Volume 6, Number 2 (1988),
I. Androulidakis, Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel, J. Math. Phys. 45, 3995 (2004); (pdf)
The automorphism groups of principal bundles are discussed for instance in