internal homotopy ∞-groupoid?
(…)
In a locally contractible (∞,1)-topos $\mathbf{H}$ 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.
If $\mathbf{H}$ is a smooth (∞,1)-topos, then nonabelian deRham cocycles are represented by flat? ∞-Lie algebroid valued differential forms $\omega$:
If $A = \mathbf{B}^n R/Z$ then $\omega$ is an ordinary closed n-form.
A differential 1-form $A \in \Omega^1(X)$ on a smooth manifold $X$ may be thought of as a connection on the trivial $U(1)$- or $\mathbb{R}$-principal bundle on $X$.
Similarly a differential 2-form $B \in \Omega^2(X)$ on a manifold $X$ may be thought of as a connection on the trivial $U(1)$-bundle gerbe on $X$; or on the trivial $\mathbf{B}(1)$-principal 2-bundle.
This pattern continues: a differential $n$-form is the same as a connection on a trivial $\mathbf{B}^n U(1)$-principal ∞-bundle.
Moreover this pattern generalizes to $G$-principal bundles for nonabelian groups $G$:
for $\mathfrak{g}$ the Lie algebra of a Lie group $G$ – possibly nonabelian – a Lie-algebra valued 1-form $A \in \Omega^1(X,g)$ may be thought of as a connection on the trivial $G$-principal bundle on $X$.
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 $X$ this is simply given by a morphism $\Pi(X) \to A$ from the path ∞-groupoid to the given coefficient object $A$.
The underlying principal ∞-bundle is that characterized by the cocycle that is given by the composite morphism $X \to \Pi(X) \to A$.
We may therefore characterize flat connections on trivial $A$-principal ∞-bundles as those morphisms $\Pi(X) \to A$ for which the composite $X \to \Pi(X) \to A$ trivializes. This way we characterize $A$-valued deRham cohomology in the (∞,1)-topos $\mathbf{H}$.
Fix a model for the (∞,1)-topos $\mathbf{H}$ in terms of the local model structure on simplicial presheaves $SPSh(C)^{loc}$ as described at path ∞-groupoid.
For $A \in SPSh(C)$ a pointed object with point $pt_A : {*} \to A$ define $A_{dR} \in SPSh(C)$ by
This we call the de Rham differential refinement of $A$.
The cohomology with coefficients in $A_{dR}$
we call $A$-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 $\mathbf{H}$ does happen to be a smooth (∞,1)-topos then both notions are compatible.