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 bunde 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 $P \to G$ 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 $\mathcal{G}$ a Lie groupoid, a $\mathcal{G}$-prinipal bundle is locally of the form
for $\mathcal{G}_{x_i}$ the source fiber over an object $x_i$.
principal bundle / torsor / groupoid principal bundle