(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
An associated $\infty$-bundle $E \to X$ is a fiber bundle in an (∞,1)-topos $\mathbf{H}$ with typical fiber $F \in \mathbf{H}$ that is classified by a cocycle $X \to \mathbf{B}\underline{Aut}(F)$ with coefficients in the delooping of the internal automorphism ∞-group of $F$. We say this is associated to the corresponding $\underline{Aut}(F)$-principal ∞-bundle.
More generally there should be notions of associated $\infty$-bundles whose fibers are objects in an (∞,n)-topos over $\mathbf{H}$ for some $n \gt 1$.
Let $\mathbf{H}$ be an (∞,1)-topos.
For $V,X \in \mathbf{H}$ two objects, say a $V$-fiber ∞-bundle over $X$ is a morphism $E \to X$ (an object in the slice (∞,1)-topos $\mathbf{H}_{/X}$) such that there exists an effective epimorphism $U \to X$ and an (∞,1)-pullback square
Let $G \in Grp(\mathbf{H})$ be an ∞-group equipped with an ∞-action $\rho$ on $V$. Then for $P \to X$ a $G$-principal ∞-bundle over $X$, the $\rho$-associated $\infty$-bundle is
where $P \times_G V := (P \times V)//G$ is the homotopy quotient of the diagonal $G$-action.
Below in Properties we see that every $\rho$-associated $\infty$-bundle is a $V$-fiber $\infty$-bundle and that every $V$-fiber $\infty$-bundle arises as associated to an $\mathbf{Aut}(V)$-principal ∞-bundle
For $V \in \mathbf{H}$, write $\mathbf{Aut}(V) \in Grp(\mathbf{H})$ for the internal automorphism ∞-group of $V$. This comes with a canonical action on $V$. Then the operation of sending an $\mathbf{Aut}$-principal ∞-bundle $P \to X$ to the associated $P \times_G V \to X$ establishes an equivalence
More specifically, if $\rho$ is an ∞-action of $G$ on some $V \in \mathbf{H}$, then under the equivalence of (∞,1)-categories
it corresponds to a fiber sequence
in $\mathbf{H}$. This is the universal $\rho$-associated $V$-bundle in that for $P \to X$ any $G$-principal ∞-bundle modulated by $g \colon X \to \mathbf{B}G$ we have a natural equivalence
This is discussed in (NSS, section I 4.1).
In (Wendt), section 5.5, a presentation of the general situation for 1-localic (∞,1)-toposes is given in terms of the model structure on simplicial presheaves (as discussed at models for ∞-stack (∞,1)-toposes) .
Under this presentation we have:
The universal $F$-$\infty$-bundle $\mathbf{E} F \to \mathbf{B}Aut(F)$ is presented by the bar construction
Compare universal principal ∞-bundle.
For the special case that $\mathbf{H} =$ ∞Grpd and using the presentation by the model structure on topological spaces/model structure on simplicial sets the classification theorem 1 reduces to the classical statement of (Stasheff, May).
In the case that the fiber $F$ is the delooping $F = \mathbf{B}G$ of an ∞-group object $G$, the $\underline{Aut}(\mathbf{B}G)$-associated $\infty$-bundles are called $G$-∞-gerbes. See there for more details.
principal ∞-bundle / ∞-gerbe / associated $\infty$-bundle
Early work on associated $\infty$-bundles takes place in the $(\infty,1)$-topos ∞Grpd $\simeq$ Top. In
a classification of fibrations of CW-complexes with given CW-complex fiber in terms of maps into a classifying CW-complex is given.
In
the total space of the universal $F$-fiber ∞-bundle in the pointed context is identified with $\mathbf{B}Aut_*(F)$ (the pointed automorphism ∞-group).
A generalization or more systematic account of the classification theory is then given in
This has been reproven in various guises, such as the statement of univalence in the model sSet for homotopy type theory. See the references at univalence for more on this.
Generalizations with extra structure on the fibers are discussed in
Consideration of associated $\infty$-bundles / fiber sequences in general 1-localic (∞,1)-toposes presented by a model structure on simplicial presheaves (which subsumes the above case for the trivial site) is discussed in
Related discussion on the behaviour of fiber sequences under left Bousfield localization of model categories is in
Similar considerations and results are in
With the advent of (∞,1)-topos theory all these statements and their generalizations follow from the existence of object classifiers in an (∞,1)-topos. For the classical case in ∞Grpd $\simeq$ Top${}^\circ$ sSet${}^\circ$ this is discussed in
which reproduces the classical results (Stasheff, May).
For general (∞,1)-toposes the classification of associated $\infty$-bundles is discussed in section I 4.1 of
Models in rational homotopy theory of classifying spaces for homotopy types $Aut(F)$ go back to Sullivan’s remarks on the automorphism L-infinity algebra. Further developments are reviewed and developed in