Schreiber de Rham cohomology in an (∞,1)-topos

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 $\omega$:

$$\array{ \mathbf{\Pi}_{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \mathbf{\Pi}(X) &\to& A } \,.$$

If $A = \mathbf{B}^n R/Z$ then $\omega$ is an ordinary closed n-form.

For more see differential cohomology in an (∞,1)-topos.

Idea

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 $\mathbf{H}$.

Definition

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.

Definition (deRham differential refinement)

For $A \in SPSh(C)$ a pointed object with point $pt_A : {*} \to A$ define $A_{dR} \in SPSh(C)$ by

$$A_{dR} : U \mapsto [I,SPSh(C)] \left( \array{ U \\ \downarrow \\ \Pi(U) } \,,\; \array{ {*} \\ \downarrow^{\mathrlap{pt_A}} \\ A } \right) \,.$$

This we call the de Rham differential refinement of $A$.

The cohomology with coefficients in $A_{dR}$

$$H_{dR}(X,A) := \pi_0 \mathbf{H}(X,A_{dR})$$

we call $A$-valued de Rham cohomology

Remark

(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.