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nonabelian de Rham cohomology > history

structures in an (∞,1)-topos

• shape

• cohomology

  • cocycle/characteristic class

  • twisted cohomology

  • principal ∞-bundle

  • ∞-vector bundle

• homotopy

  • covering ∞-bundles

  • Postnikov system

  • path ∞-groupoid

  • geometric realization

  • Galois theory

  • internal homotopy ∞-groupoid?

  • Whitehead system

• rational homotopy

  • ∞-Lie algebroid

  • ordinary rational homotopy

  • internal rational homotopy

  • Chern-character

• differential cohomology

  • flat differential cohomology

  • de Rham cohomology

  • de Rham theorem

• relative theory over a base

  • relative homotopy theory

  • Lie theory

Examples

(…)

Applications

• Background fields in twisted differential nonabelian cohomology

• Differential twisted String and Fivebrane structures

• D'Auria-Fre formulation of supergravity

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In a locally contractible (∞,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.

If $H$ is a smooth (∞,1)-topos, then nonabelian deRham cocycles are represented by flat? ∞-Lie algebroid valued differential forms $\omega$:

$$\begin{array}{ccc}{\Pi }^{\inf }(X)& \stackrel{\omega }{\to }& 𝔞\\ \downarrow & & \downarrow \\ \Pi (X)& \to & A\end{array}.$$\array{ \Pi^{inf}(X) &\stackrel{\omega}{\to}& \mathfrak{a} \\ \downarrow && \downarrow \\ \Pi(X) &\to& A } \,.

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

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 $ℝ$-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 $B(1)$-principal 2-bundle.

This pattern continues: a differential $n$-form is the same as a connection on a trivial $B^{n}U(1)$-principal ∞-bundle.

Moreover this pattern generalizes to $G$-principal bundles for nonabelian groups $G$:

for $𝔤$ 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 $H$.

Definition

Fix a model for the (∞,1)-topos $H$ in terms of the local model structure on simplicial presheaves $\mathrm{SPSh}(C{)}^{\mathrm{loc}}$ as described at path ∞-groupoid.

Definition (deRham differential refinement)

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

$$A_{\mathrm{dR}}:U\mapsto [I,\mathrm{SPSh}(C)](\begin{array}{c}U\\ \downarrow \\ \Pi (U)\end{array},\begin{array}{c}*\\ {\downarrow }^{{\mathrm{pt}}_A}\\ A\end{array}).$$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 $A_{\mathrm{dR}}$

$$H_{\mathrm{dR}}(X,A):={\pi }_{0}H(X,A_{\mathrm{dR}})$$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 $H$ does happen to be a smooth (∞,1)-topos then both notions are compatible.