In a locally contractible (∞,1)-topos a cocycle in (nonabelian) de Rham cohomology is a cocycle in flat differential cohomology whose underlying cocycle in (nonabelian) cohomology is trivial: it encodes a trivial principal ∞-bundle with possibly nontrivial but flat connection.
If then is an ordinary closed n-form.
This pattern continues: a differential -form is the same as a connection on a trivial -principal ∞-bundle.
Moreover this pattern generalizes to -principal bundles for nonabelian groups :
While it may seem that the notion of differential form is more fundamental than that of a connection, in the context of differential nonabelian cohomology in an arbitrary path-structured (∞,1)-topos? the most fundamental notion of a differential cocycle is that of a flat connection on a principal ∞-bundle : on an space this is simply given by a morphism from the path ∞-groupoid to the given coefficient object .
We may therefore characterize flat connections on trivial -principal ∞-bundles as those morphisms for which the composite trivializes. This way we characterize -valued deRham cohomology in the (∞,1)-topos .
For a pointed object with point define by
This we call the de Rham differential refinement of .
The cohomology with coefficients in
we call -valued de Rham cohomology
(de Rham cohomology in terms of differential forms)
The definition does not actually presuppose that the ambient (∞,1)-topos is a smooth (∞,1)-topos in which a concrete notion of ∞-Lie algebroid valued differential forms exists. It defines a notion of “de Rham cohomology” even in the absence of an ordinary notion of differential forms.
But if does happen to be a smooth (∞,1)-topos then both notions are compatible.