(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The notion of principal $\infty$-bundle is a categorification of principal bundle from groups and groupoids to ∞-groupoids, or rather from parameterized groupoids (generalized spaces called stacks) to parameterized $\infty$-groupoids (generalized spaces called ∞-stacks).
For motivation, background and further details see
A model for principal $\infty$-bundles is given by
See also
We define $G$-principal $\infty$-bundles in the general context of an ∞-stack (∞,1)-topos $\mathbf{H}$, with $G$ a group object in the (∞,1)-topos.
Recall that for $A \in \mathbf{H}$ an object equipped with a point $pt_A : {*} \to A$, its corresponding loop space object $\Omega A$ is the homotopy pullback
Conversely, for $G \in \mathbf{H}$ we say an object $\mathbf{B}G$ is a delooping of $G$ if it has an essentially unique point and if $G \simeq \Omega \mathbf{B}G$. We call $G$ an ∞-group. More in detail, its structure as a group object in an (∞,1)-category is exhibited by the Čech nerve
of ${*} \to \mathbf{B}G$.
To every cocycle $g : X \to \mathbf{B}G$ is canonically associated its homotopy fiber $P \to X$, the (∞,1)-pullback
We discuss now that $P$ canonically has the structure of a $G$-principal ∞-bundle and that $\mathbf{B}G$ is the fine moduli space for $G$-principal $\infty$-bundles.
(principal $G$-action)
Let $G$ be a group object in the (∞,1)-topos $\mathbf{H}$. A principal action of $G$ on a morphism $(P \to X) \in \mathbf{H}$ is a groupoid object $P//G$ that sits over $*//G$ in that we have a morphism of simplicial diagrams $\Delta^{op} \to \mathbf{H}$
in $\mathbf{H}$;
and such that $P \to X$ exhibits the (∞,1)-colimit
called the base space over which the action takes place.
We may think of $P//G$ as the action groupoid of the $G$-action on $P$. For us it defines this $G$-action.
The $G$-principal action as defined above satisfies the principality condition in that we have an equivalence of groupoid objects
This principality condition asserts that the groupoid object $P//G$ is effective. By Giraud's axioms characterizing (∞,1)-toposes, every groupoid object in $\mathbf{H}$ is effective.
For $X \to \mathbf{B}G$ any morphism, its homotopy fiber $P \to X$ is canonically equipped with a principal $G$-action with base space $X$.
First we show that we have a morphism of simplicial diagrams
with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.
The defining (∞,1)-pullback square for $P \times_X P$ is
To compute this, we may attach the defining $(\infty,1)$-pullback square of $P$ to obtain the pasting diagram
and use the pasting law for pullbacks, to conclude that $P \times_X P$ is the pullback
By definition of $P$ as the homotopy fiber of $X \to \mathbf{B}G$, the lower horizontal morphism is equivbalent to $P \to {*} \to \mathbf{B}G$, so that $P \times_X P$ is also equivalent to the pullback
This finally may be computed as the pasting of two pullbacks
of which the one on the right is the defining one for $G$ and the remaining one on the left is just an (∞,1)-product.
Proceeding by induction from this case we find analogously that $P^{\times_X^{n+1}} \simeq P \times G^{\times_n}$: suppose this has been shown for $(n-1)$, then the defining pullback square for $P^{\times_X^{n+1}}$ is
We may again paste this to obtain
and use from the previous induction step that
to conclude the induction step with the same arguments as before.
This shows that $P//G$ is the Cech nerve of $P \to X$. It remains to show that indeed $X = {\lim_\to}_n P \times G^{\times^n}$. For this notice that $* \to \mathbf{B}G$ is an effective epimorphism in an (infinity,1)-category. Hence so is $P \to X$. This proves the claim, by definition of effective epimorphism.
using this we have
We have established that every cocycle $X \to \mathbf{B}G$ canonically induced a $G$-principal action on the homotopy fiber $P \to X$. The following definition declares the $G$-principal $\infty$-bundles to be those $G$-principal actions that do arise this way.
We say a $G$-principal action of $G$ on $P$ over $X$ is a $G$-principal ∞-bundle if the colimit over $P//G \to *//G$ produces a pullback square: the bottom square in
For $G$ an infinity-group in $\mathbf{H}$ and $X \in \mathbf{H}$ any object, write
for the sub-(infinity,1)-category on the over-(infinity,1)-category of the groupoid objects over $*//G$ on the $G$-principal $\infty$-bundles as above.
We have an equivalence of (∞,1)-categories
of $G$-orincipal $\infty$-bundles over $X$ with cocycles $X \to \mathbf{B}G$.
The arrow category $\mathbf{H}^I$ is still an (infinity,1)-topos and hence the Griraud-Lurie axioms still hold. This means that by the discussion at groupoid object in an (infinity,1)-category (using the statement below HTT, cor. 6.2.3.5) we have an equivalence
between groupoid objects in $\mathbf{H}^I$ and effective epimorphisms in the arrow category.
Notice that groupoid objects and effective epis in $\mathbf{H}^I$ are given objectwise over the two objects of the inerval $I = \Delta[1]$.
Restricting this equivalence along the inclusion
given by sending a cocycle to its homotopy fiber diagram
therefore yields precisely the equivalence in question
In words this says that the cohomology on $X$ with coefficients in $G$ classified $G$-principal $\infty$-bundles, and in fact does so on the level of cocycles.
For some comments on the generalization of the notion of connection on a bundle to principal $\infty$-bundles see differential cohomology in an (∞,1)-topos -- survey.
We discuss realizations of the general definition in various (∞,1)-toposes $\mathbf{H}$.
The following general construction was originally due to Quillen and defines principal groupoid $\infty$-bundles in the (∞,1)-topos Top in its presentation by the model structure on simplicial sets.
Let $C$ be a small category and let
be a functor with values in SSet such that it sends all morphisms in $C$ to weak equivalences in SSet (weak homotopy equivalences of simplicial sets).
Consider first the case that $C$ has a single object, so that it is the delooping $\mathbf{B}G$ of a monoid or group $G$. Then
Let
be the simplicial set assigned to this single object and let
be the corresponding action groupoid (see there for the description as a weak colimit).
Notice that, as every action group, this comes with a canonical map $P//G \to \mathbf{B}G$.
Given the above, the diagram
is a homotopy pullback (i.e. defines a fibration sequence).
This is originally due to
The statement is reproduced in section IV of
For the simple case that $G$ is group, in which case $\rho_C$ necessarily takes values not just in weak equivalences but is isomorphisms of simplicial sets, this says that $P \to X$ is a $G$-principal $\infty$-bundle. In particular the principality of the action is manifestly exhibited by the fact that the base space $X$ is the (weak) quotient of $P$ by the action of $G$.
The above reproduces manifest the description of ordinary $G$-principal topological bundles in the incarnation as groupoids as described in detail at generalized universal bundle.
More generally, when $G$ is just a monoid the above descibes something a bit more general than an ordinary $G$-principal bundle (as then the action of $G$ on the total space may be by weak equivalences that are not isomorphisms).
Quillen’s original construction is more general than this, concerning in fact 1-groupoid-principal $\infty$-bundles:
Let now $C$ be a category and for
a functor that sends all morphisms to weak equivalences of simplicial sets.
Let now for each object $c \in C$
be the “bundle of $c$-fibers”.
Then for each $c$ the diagram
is a homotopy pullback (i.e. defines a fibration sequence).
This classical construction is recalled in the introduction of
See simplicial principal bundle.
For $X$ a topological space $C = Op(X)$ the category of open subsets of $X$, let $\mathbf{H} = Sh_{(\infty,1)}(X)$ be the (∞,1)-topos of ∞-stacks on $C$. This is the petit topos incarnation of $X$.
In its presentation by the model structure on simplicial presheaves this is the context in which princpal $\infty$-bundles are discussed in
For $C$ a site of test space, – for instance duals of algebras over a Lawvere theory as described at function algebras on infinity-stacks – let $\mathbf{H} = Sh_{(\infty,1)}(C)$ be the (∞,1)-topos of ∞-stacks on $C$. This is a gros topos.
Smooth principal $\infty$-bundles are realized in the $\infty$-Cahiers topos as described in some detail at ∞-Lie groupoid.
In this context there is a notion of connection on a principal ∞-bundle.
For $G$ an ordinary Lie group, a $G$-principal bundle in the $(\infty,1)$-topos $\mathbf{H} =$ ∞LieGrpd is an ordinary $G$-principal bundle.
For $G = \mathbf{B}^{n-1} U(1) \in$ ∞LieGrpd, the circle Lie n-group, a $G$-principal $\infty$-bundle is a circle $n$-bundle.
See circle n-bundle with connection.
Classes of examples include
A bundle gerbe is a concrete model for the total space groupoid of the total space of a $\mathbf{B}U(1)$-principal 2-bundle.
More generally, a nonabelian bundle gerbe is a concrete model for the groupoid of the total space of a general principal 2-bundle.
A bundle 2-gerbe is a concrete model for the total space 2-groupoid of the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle.
More generally, a nonabelian bundle 2-gerbe is a concrete model for the 2-groupoid of the total space of a general principal 3-bundle.
Classes of examples include
A principal $\infty$-bundle over a 0-connected object / delooping object $\mathf{B}K$ is a normal morphism of ∞-groups. See there for more details.
principal bundle / torsor / associated bundle / twisted bundle
principal $\infty$-bundle / associated ∞-bundle / ∞-gerbe, twisted ∞-bundle, groupoid-principal ∞-bundle
The notion of principal $\infty$-bundle (often addressed in the relevant literature in the language of torsors) appears in the context of the simplicial presheaf model for generalized spaces in
Rick Jardine, Luo, Higher order principal bundles (pdf).
Rick Jardine, Cocycle categories (pdf).
An earlier description in terms of simplicial objects is
In that article not the total space of the bundle $P \to X$ is axiomatized, but the $\infty$-action groupoid of the action of $G$ on it.
See the remarks at principal 2-bundle.
See also
on associated ∞-bundles.
The fully general abstract formalization in (∞,1)-topos theory as discussed here was first indicated in
A more comprehensive conceptual account is in
The classifying spaces for a large class of principal $\infty$-bundles are discussed in
A fairly comprehensive account of the literature is also in the introduction of NSS 12, “Presentations”.
For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-infinity-actions is maybe due to
This is mentioned for instance as exercise 4.2in
Closely related discussion of homotopy fiber sequences and homotopy action but in terms of Segal spaces is in section 5 of
There, conditions are given for a morphism $A_\bullet \to B_\bullet$ to a reduced Segal space to have a fixed homotopy fiber, and hence encode an action of the loop group of $B$ on that fiber.