(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $G$ a group (internal to some category, traditionally that of topological spaces) and $X$ some other object, a $G$-principal bundle over $X$ – also called a $G$-torsor over $X$ – is a bundle $P \to X$ equipped with a $G$-action $\rho : P \times G \to P$ on $P$ over $X$, 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 $G$-principal bundles over $X$ is that they are a geometric model of the degree-1 nonabelian cohomology of $X$ with coefficients in $G$. More precisely (subject to some technical details discussed below) there is a natural isomorphism
between the degree-1 $G$-cohomology of $X$ and the isomorphism classes of $G$-principal bundles over $X$.
The “naturality” of this relation is more pronounced when one refines it from cohomology to cocycle groupoids. This is discussed below in some section,
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
Finally in
we discuss how the traditional setup and many other contexts are recovered from and illuminated by that abstract perspective.
We discuss here principal bundles in the context Top of topological spaces. So the group $G$ here is a topological group.
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 $G$ be a topological group.
The trivial $G$-principal bundle on a topological space $X$ is the product space $X \times G$ equipped with
the projection map $p_1 : X \times G \to X$ ;
the action of $G$ on $X \times G$ by right multiplication of $G$ on itself.
A $G$-principal bundle over a topological space $X$ is a topological space $P$ equipped with
a continuous function $p : P \to X$;
an action $\rho : P \times G \to P$ of $G$ on $P$ over $X$, hence fitting into a coequalizing diagram
such that this is locally trivial in the sense that
In the references listed below, this appears for instance as (Mitchell, section 2, …)
A central property of the above definition of principal bundle is
For $P \to X$ a $G$-principal bundle, it is naturally isomorphic to the quotient projection $P \to P/G \simeq X$ of the $G$-action.
(Cartan principal bundles)
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 61, Def. 1.1.2) and refer to this alternative definition of principal bundle as that of Cartan principal bundle:
Let $G$ be a locally compact topological group and $P$ a completely regular topological space equipped with a continuous function action $\rho : P \times G \to P$. If $G$ acts freely on $P$, (no element $g \in G$ except the neutral element has any fixed points in $P$ under the action) then the coprojection
to the topological quotient is called a $G$-principal bundle in the wide sense. If furthermore the division map
is a continuous function, then this is called a Cartan principal bundle (Palais 61, around theorem 1.1.3), following (Cartan).
(…)
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 $\mathbf{H}$. This we discuss now, and then relate it to the traditional notion and to various other generalizations. More along these lines is at geometry of physics – principal bundles.
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
where on the left we have the groupoid of cocycles with coefficients in the internal delooping $\mathbf{B}G$ of the group object $G$: the moduli stack of $G$-principal bundles.
In this context, all of the non-natural aspects of the traditional theory of principal bundles disappear, for instance
every $G$-principal bundle is locally trivial in a $(2,1)$-topos $\mathbf{H}$;
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.
Moreover, all these fact are fairly direct consequences simply of the Giraud axioms that characterize (2,1)-toposes in the first place.
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 $G$-action on some object $P$ is not suitably free, then the $(2,1)$-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 $G$-action on the terminal object $*$ becomes principal over the action groupoid $*//G$ and the resulting $G$-principal bundle is nothing but the universal one.
As the notation suggests, thus formulating the theory in (2,1)-topos theory immediately generalizes it to (∞,1)-topos theory. This is discussed at principal ∞-bundle.
(…)
The following is old material collected from elsewhere that is going to be rearranged….
This indicates the more fundamental way to define $G$-principal bundles in the first place:
Recall (from fiber sequence) that for every group there is the one-object groupoid $\mathbf{B}G$. Under the Yoneda embedding this represents a prestack. Write $\bar{\mathbf{B}G}$ for the corresponding stack obtained by stackification. This is our $G Bund(-)$
This perspective in turn is by general abstract nonsense equivalent to the following useful description:
Let $H$ be the suitable (∞,1)-topos internal to which one looks at $G$-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 $G Bund(X) \simeq \bar{\mathbf{B} G}(X)$ is given by a morphism in $H(X,\mathbf{B}G)$, which may be thought of as an anafunctor to $\mathbf{B}G$ from the (categorially) discrete category $X$; the $G$-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 $X \to \mathbf{B}G$:
This diagram, incidentally, directly tells us about another important property of $G$-principal bundles: they all canonically trivialize when pulled back to their own total space $P$.
This is what the homotopy commutativity of the above homotopy pullback diagram says: the cocycle $X \to \mathbf{B}G$ pulled back to the bundle $P \to X$ that it classifies becomes $P \to X \to \mathbf{B}G$, which is homotopic to the trivial cocycle (the one that factors through the point) on $P$.
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 ${*} \to \mathbf{B}G$ by $\mathbf{E}G \to \mathbf{B}G$, with $\mathbf{E}G$ an object weakly equivalent to the point, called the $G$-universal principal bundle.
With that the above homotopy pullback is computed as the ordinary pullback
So every $G$-principal bundle $P \to X$ is the pullback along a classifying map $X \to \mathbf{B}G$ (in the right $(\infty,1)$-categorical context, otherwise a span such as an anafunctor) of the $G$-universal principal bundle.
Given the definition of the bundle $P$ in terms of a homotopy pullback of ${*} \to \mathbf{B}G$ we re-obtain the $G$-action on $P$ as follows (with an eye towards its generalization to principal ∞-bundles).
Let
be the effective groupoid object in an (∞,1)-category that exhibits the delooping $\mathbf{B}G$ of $G$.
Form the homotopy pullback of the classifying morphism $X \to \mathbf{B}G$ along the $d_0$-face maps of this diagram. This yields a diagram
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 $G$ on $P$ in that the action is the morphism indicated $\rho$ in
and it exhibits $P \times G^{\times (n-1)}$ as the groupoid object in an (∞,1)-category being the ?ech nerve? of $P \to X$:
Here the second statement in particular encodes the familiar way to formulate principality of the action $\rho$, 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 $G$ on $P$.
We now rederive the action $\rho$ of $G$ on $P$ given just the classifying map $X \to \mathbf{B} G$ 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 $X \to \mathbf{B}G$ is represented by an anafunctor, namely a cocycle
in ?ech cohomology coming from some cover $\{U_i \to X\}$ of $X$.
The ?ech nerve? $C(U)$ has
objects = $\{(x,i) | x \in U_i\}$
morphisms = $\{ (x,i) \stackrel{}{\longrightarrow} (x,j) | x \in U_{i j}\}$ .
The functor $g : C(U) \to \mathbf{B}G$ sends
for $g_{i j} \in Functions(U_{i j}, G)$ as described in detail at ?ech cohomology.
With $\mathbf{E}G = \{g \stackrel{h}{\to} g h | g,h \in G \}$ the fibrant replacement of the point, which we shall find it helpful to think of as given by
objects = $\left\{ \array{ && \bullet \\ & {}^g\swarrow \\ \bullet } \right\}$
morphisms = $\left\{ \array{ && \bullet \\ & {}^g\swarrow && \searrow^{g' = g h} \\ \bullet &&\stackrel{h}{\longrightarrow}&& \bullet } \right\}$
we compute the homotopy pullback as the homotopy fiber product given by the ordinary pullback (see category of fibrant objects for details)
So we read off that $P$ is the groupoid with
objects =
morphisms =
With $P$ 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 $\eta_P$ here is
At the same time recall from the discussion at delooping that the component of the transformation $\eta_G$ in
is
Taken together this shows that the universal morphism $P \times G \to P$ 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 $\rho : P \times G \to G$ of $G$ on $P$.
We discuss here aspects of formulating a theory of principal bundles in contexts different from those already discussed above.
(…)
In higher category theory the notion of principal bundle has various vertical categorifications. See
For $P \to X$ a $G$-principal bundle, its Atiyah Lie groupoid is
with the evident composition operation.
The principal bundle $P \to X$ 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).
A standard example of a nontrivial circle group-principal bundle – a circle bundle – is the Hopf fibration $S^3 \to S^2$, which has the structure of an $S^1$-principal bundle in topological spaces.
Generally, if we accept that we have a large supply of continuous maps between topological spaces, we obtain a $G$-principal bundle $f^* \mathcal{E}G \to X$ on a space $X$ for each continuous map $f : X \to \mathcal{B}G$ to the classifying space of $G$, by pullback of the universal bundle $\mathcal{E}G \to \mathcal{B}G$ along $f$.
We consider actions by topological groups and Lie groups.
For $X$ a smooth manifold and $G$ a compact Lie group equipped with a free smooth action on $X$, then the quotient projection
is a $G$-principal bundle (hence in particular a Serre fibration).
This is originally due to (Gleason 50). See e.g. (Cohen, theorem 1.3)
Let $P$ be a completely regular topological space and let $G$ be a Lie group equipped with a free action on $P$. Then the quotient map $P \to P/G$ is a $G$-principal bundle – in that it is locally trivial – precisely if the division map
is a continuous function.
This is (Palais, theorem 4.1).
We discuss principal bundles of the form $P \to P/G$ for $G \hookrightarrow P$ a subgroup of a topological group, hence with base space a coset space.
For $G$ a Lie group and $H \subset G$ a compact subgroup, then the coset quotient projection
is an $H$-principal bundle (hence in particular a Serre fibration).
This is a direct corollary of prop. . Originally this statement is due to (Samelson 41).
For $G$ a compact Lie group and $K \subset H \subset G$ closed subgroups, then the projection map
is a locally trivial $H/K$-fiber bundle (hence in particular a Serre fibration).
Observe that the projection map in question is equivalently
(where on the left we form the Cartesian product and then divide out the diagonal action by $H$). This exhibits it as the $H/K$-fiber bundle associated to the $H$-principal bundle of corollary .
If $P$ is a topological group and $G \hookrightarrow P$ a closed Lie subgroup, then the quotient map $P \to P/G$ is a locally trivial $G$-principal bundle.
This is a corollary of theorem (Palais 61).
Examples where $P \to P/G$ is not locally trivial are in (Karube), see also (Mostert):
Let $P$ be the product of infinitely many circles, and let $G$ be the product of their order 2 subgroups. This cannot have local section because $P$ is locally connected and $G$ is not. Therefore $P$ is not even locally homeomorphic to $(P / G) \times G$.
In physics, principal bundles with connection and their higher categorical analogs model gauge fields. See at fiber bundles in physics.
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 $U(1)$-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.
(formally) principal bundle / (pseudo-)torsor / groupoid principal bundle / associated bundle
gauge field: models and components
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 61).
See also the references at classifying space.
Textbook accounts:
Norman Steenrod, section I.7 of The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)
Dale Husemöller, Fiber bundles, Springer (1994) (doi:10.1007/978-1-4757-2261-1)
Dale Husemoeller, Michael Joachim, Branislav Jurco, Martin Schottenloher, Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, Springer 2008 (pdf)
Ralph Cohen, The topology of fiber bundles, Stanford University (2017) (pdf, OMN:201707.110706)
Loring Tu, Parts I-II in: Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)
With an eye towards application in mathematical physics:
Mikio Nakahara, Chapter 9 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)
Gerd Rudolph, Matthias Schmidt, Section 1.1: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)
For principal bundles in the smooth context see most textbooks on differential geometry, for instance
also around section 3.1 of
Questions related to the existence slices of topological G-spaces, of sections of $G$-bundles and conditions for properness of some related maps are treated in
Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 (jstor:1970335, doi:10.2307/1970335, pdf)
Lecture notes on principal bundles include
Anders Kock, Fibre bundles in general categories, Journal of Pure and Applied Algebra 56(3):233-245, 1989; Generalized fibre bundles, in: Categorical Algebra and its Applications, Lecture Notes in Mathematics 1348, pp 194-207 (2006)
C. Townsend, Principal bundles as Frobenius adjunctions with application to geometric morphisms, Math. Proc. Camb. Phil. Soc. 159(03) (2015), 433-444 pdf
Tomasz Brzezinski, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213
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
Discussion of topological quotients of groups $G \to G/H$ as principal $H$-bundles is in
Andrew Gleason, Spaces with a compact Lie group of transformations, Proc. of A.M.S 1, (1950), 35 - 43.
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
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