(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
For $\mathcal{G}$ a groupoid object a $\mathcal{G}$-principal bundle is a morphism $P \to X$ with an principal action of $\mathcal{G}$ on $P$.
For $G$ a group object in some (∞,1)-topos $\mathbf{H}$ (for instance $\mathbf{H} =$ ?LieGrpd? for smooth Lie groupoid-bundles), and $\mathbf{B}G$ the corresponding delooping object, $G$-principal bundles are the (∞,1)-pullbacks of the form
One equivalently (with non-negligible but conventional chance of confusion of terminology) calls such $P \to X$ a $\mathbf{B}G$-groupoid principal bundle.
So more generally, for $\mathcal{G}$ any groupoid object with collection $\mathcal{G}_0$ of objects, the $(\infty,1)$-pullbacks
are groupoid principal bundles .
For $\mathbf{H}$ = $\infty LieGrpd$ and $G:=G_1 \Rightarrow G_0$ a Lie groupoid, a $G$-principal bundle is locally of the form
for $\mathcal{G}_{x_i}$ the target fiber over an object $x_i$, or equivalently of the form $U_i\times_{G_0} G_1$.
More precisely, we have the following sub-definition for Lie groupoid principal bundles: Given a Lie groupoid $G=G_1 \Rightarrow G_0$, a $G$-principal bundle over a manifold $M$ (with right $G$ action) is a surjective submersion $P\xrightarrow{\pi} M$ together with a moment map $P\xrightarrow{\rho} G_0$ (sometimes also called an anchor), such that there is a (right) $G$-action on $P$, that is, there is map $\Phi: P\times_{G_0} G_1 \to P$, such that
$\pi(\Phi(p, g))=\pi(p)$,
$\rho(\Phi(p,g))=s(g)$,
$\Phi(\Phi(p, g_1), g_2)=\Phi(p, g_1 g_2)$.
Moreover, we require the $G$-action to be free and proper, that is,
[free] $P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is injective,
[proper]$P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is a proper map , i.e. the preimage of a compact is compact.
These two conditions are equivalent to the fact that
is an isomorphism.
principal bundle / torsor / groupoid principal bundle