Schreiber
nonabelian de Rham cohomology

In a ∞-connected (∞,1)-topos $H$ a cocycle in (nonabelian) de Rham cohomology is a cocycle $\Pi (X)\to A$ in flat differential cohomology whose underlying cocycle $X\hookrightarrow \Pi (X)\to A$ in (nonabelian) cohomology is trivial: it encodes a trivial principal ∞-bundle with possibly nontrivial but flat connection.

Details are at

• differential cohomology in an (∞,1)-topos in the section Intrinsic de Rham cohomology.

See also ∞-Lie algebroid valued differential forms.