(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
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
For $G$ a model for an ∞-group, there is often a model for the universal principal ∞-bundle $\mathbf{E}G$ that itself carries a group structure such that the canonical inclusion $G \to \mathbf{E}G$ is a homomorphism of group objects. This extra groupal structure is important for various constructions.
For $G$ an ordinary bare group, the action groupoid $\mathbf{E}G = G//G$ of the right multiplcation action of $G$ on itself
is contractible. We may think of this as the groupoid $INN(G) \subset AUT(G)$ of inner automorphisms inside the automorphism 2-group of $G$. But it is also simply isomorphic to the codiscrete groupoid on the set underlying $G$. In this latter form the 2-group structure on $\mathbf{E}G$ is manifest, which corresponds to the crossed module $(G \stackrel{Id}{\to} G)$ .
It is also manifest then that this fits with the one-object delooping groupoid $\mathbf{B}G$ of $G$ into a sequence
where the first morphism is a homomorphism of strict 2-groups.
Regarded as a sequence of morphisms in the model KanCplx of the (∞,1)-topos ∞-Grpd this is already a model for the universal $G$-bundle.
If $G$ here is refined to a Lie group or topological group then this is a sequence of ∞-Lie groupoids or topological ∞-groupoids, respectively, and also then, this is already a model for the universal $G$-principal bundle, as discussed at ∞LieGrpd – the universal G-principal bundle.
By applying the geometric realization functor
we obtain a sequence of topological spaces
where $\mathcal{E}G$ carries the structure of a topological group and the morphism $G \to \mathcal{E}G$ is a topological group homomorphism. For bare groups $G$ and under mild assumptions also for general topological groups $G$, this groupal topological model for the universal $G$-bundle obtained from the realization of the groupoid $G//G$ was consider in Segal68.
For $G \in$ ∞Grpd a strict 2-group a groupal model for $\mathbf{E}G$ was given in RobertsSchr07 generalizing the $INN(G) \subset AUT(G)$ construction mentioned above. This yields a weak 3-group structure on $\mathbf{E}G$ (A Gray-group).
In Roberts07 it is observed that there is also an analog of $codisc(G)$ and that this yields a strict group structure on $\mathbf{E}G$. In fact, this strictly groupal model of $\mathbf{E}G$ turns out to be isomorphic to the standard model for the universal simplicial principal bundle traditionally denoted $W G$. And this statement generalizes…
Every ∞-group may be modeled by a simplicial group $G$. There is a standard Kan complex model for the universal $G$-simplicial principal bundle $\mathbf{E}G \to \mathbf{B}G$ denoted $W G \to \bar W G$.
This standard model $W G$ does happen to have the structure of a simplicial group itself, and this structure is compatible with that of $G$ in that the canonical inclusion $G \to W G$ is a homomorphisms of simplicial groups.
This sounds like a statement that must be well known, but it was maybe not in the literature. The simplicial group structure is spelled out explicitly in Roberts07
Since the $W$ construction discussed above is functorial, this generalizes to prresheaves of simplicial groups and hence gives models for group objects and groupal universal principal ∞-bundles in (∞,1)-toposes modeled by the model structure on simplicial presheaves.
(…)
In the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids we can obtain ∞-groups by Lie integration of ∞-Lie algebras.
Corresponding to this is a construction of Lie-integrated groupal universal principal $\infty$-bundles:
for $\mathfrak{g}$ an $L_\infty$-algebra, there is an $L_\infty$-algebra $inn(\mathfrak{g})$, defined such that its Chevalley-Eilenberg algebra is the Weil algebra $W(\mathfrak{g})$ of $\mathfrak{g}$:
Under Lie integration this gives a groupal model for the universal principal $\infty$-bundle over the ∞-Lie group that integrates $\mathfrak{g}$.
This is described at
The Lie-integrated universal principal ∞-bundle
universal principal ∞-bundle , groupal model for universal principal ∞-bundle
The observation that for $G$ an ordinary group, its action groupoid sequence $G \to G//G \to \mathbf{B}G$ – which is the strict 2-group coming from the crossed module $(G \stackrel{Id}{\to} G)$ - maps under the nerve to the universal $G$-bundle appeared in
A weak 3-group structure on $G \to \mathbf{E}G$ for $G$ a strict 2-group is descibed in
The simplicial group structure on $G \to \mathbf{E}G$ for $G$ a general simplicial group is stated explicitly in
A general abstract construction of this simplicial group structure is discussed in
The use of L-∞-algebras $inn(\mathfrak{g})$ as $L_\infty$-algebraic models for universal $\mathfrak{g}$-principal bundle (evident as it is) was considered as such in