# nLab non-abelian de Rham cohomology

Contents

cohomology

### Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

For $X$ a smooth manifold, the traditional coboundary-relation which defines the ordinary de Rham cohomology-classes $[\omega] \in H^n{dR}(X)$ of closed differential n-forms $\omega \in \Omega_{dR}^n(X)_{clsd}$

$[\omega] = [\omega'] \;\;\;\; \Leftrightarrow \;\;\;\; \underset{ \alpha \in \Omega_{dR}^{n-1}(X) }{\exists} \omega' = \mathrm{d}\alpha+ \omega$

is equivalent to the concordance-relation [FSS20, Prop. 6.4]

(1)$[\omega] = [\omega'] \;\;\;\; \Leftrightarrow \;\;\;\; \underset{ \widehat{\omega} \in \Omega_{dR}^{n1}(X \times [0,1])_{clsd} }{\exists} \; \left\{ \begin{array}{l} \omega = \widehat\omega\vert_{0} \,, \\ \omega' = \widehat\omega\vert_{1} \,. \end{array} \right.$

But the latter concordance-relation immediately generalizes to flat $L_\infty$-algebra valued differential forms

$\Omega_{dR}\big( X; \mathfrak{a} \big)_{clsd} \;\; \coloneqq \;\; Hom_{dgAlg}\big( CE(\mathfrak{a}) ,\, \Omega^\bullet_{dR}(X) \big)$

with coefficients in any $L_\infty$-algebra $\mathfrak{a}$, which reduces to the ordinary case for $\mathfrak{a} \equiv b^{n-1} \mathbb{R}$ the line Lie $n$-algebra.

Therefore it makes sense to define [FSS20, Def. 6.3]:

###### Definition

The non-abelian de Rham cohomology of a smooth manifold $X$ with coefficients in a $L_\infty$-algebra $\mathfrak{a}$ is the set of concordance classes of flat $\mathfrak{a}$-valued differential forms on $X$:

(2)$H_{dR}\big( X ;\, \mathfrak{a} \big) \;\; \coloneqq \;\; \Omega_{dR}\big( X ;\, \mathfrak{a} \big) _{clsd} \big/ \mathrm{concordance} \,.$

## Examples

###### Example

With (1) it follows that the ordinary de Rham cohomology in degree $n$ is equivalently non-abelian de Rham cohomology with coefficients in the line Lie n-algebra $b^{n-1}\mathfrak{u}(1)$:

$H_{dR}\big( X ;\, b^{n-1}\mathfrak{u}(1) \big) \;\; \simeq \;\; H^n_{dR}\big( X ;\, b^{n-1}\mathbb{R} \big) \,.$

###### Example

In higher gauge theories of Maxwell-type, nonabelian de Rham cohomology of a Cauchy surface with coefficients in an L-infinity algebra characteristic of the theory’s Gauss law reflects the total flux of the higher gauge fields.

## Properties

### Recipient of non-abelian character map

For $\mathcal{A}$ (the homotopy type of) a topological space which is nilpotent (for instance: simply connected) and of rational finite type (all its rational cohomology-groups are finite-dimensional $\mathbb{Q}$-vector spaces) one may regard the homotopy classes of maps into $\mathcal{A}$ as the nonabelian cohomology classified by $\mathcal{A}$ (the non-abelian cohomology in degree=1 with coefficients in the loop space $\infty$-group $\Omega \mathcal{A}$ ):

(3)$H\big( X ;\, \mathcal{A} \big) \;\; \coloneqq \;\; \pi_0 \, Maps\big( X ,\, \mathcal{A} \big) \;\; \simeq \;\; \pi_0 \, Maps\big( X ,\, B \Omega \mathcal{A} \big) \;\; \equiv \;\; H^1\big( X ;\, \Omega \mathcal{A} \big) \,.$

For example, in the case that

$\mathcal{A} \,\equiv\, K(n,A)$

is an Eilenberg-MacLane space for a discrete abelian group $A$, this reduces to ordinary cohomology:

$H\big( X ;\, K(n,A) \big) \;\; \simeq \;\; H^n(X;\, A) \,,$

or if

$\mathcal{A} \;\equiv\; KU_0 \,\simeq\, B U \times \mathbb{Z}$

is the classifying space KU$_0$ for complex topological K-theory, then this reduces to to complex topological K-theory:

$H\big( X ;\, KU_0 \big) \;\; \simeq \;\; K(X) \,.$

Generally, if $\mathcal{E}$ is an Omega-spectrum of spaces, then

$H\big( X ;\, E_n \big) \;\; \simeq \;\; E^n(X)$

coincides with the Whitehead-generalized $E$-cohomology.

Now the rationalization-unit $\eta^{\mathbb{Q}}_{\mathcal{A}} \,\colon\, \mathcal{A} \to \mathcal{A}_{\mathbb{Q}}$ followed by suitable extension of scalars along $\mathbb{Q} \to \mathbb{R}$ induces cohomology operations in the non-abelian cohomology (3), to what may be called non-abelian rational cohomology, and non-abelian real cohomology with coefficients in $\mathcal{A}$

(4)$H\big( -;\, \mathcal{A} \big) \longrightarrow H\big( -;\, \mathcal{A}_{\mathbb{Q}} \big) \longrightarrow H\big( -;\, \mathcal{A}_{\mathbb{R}} \big) \;\; \simeq \;\; H_{dR}\big( - ;\, \mathfrak{l}\mathcal{A} \big) \,,$

and, shown on the right, a non-abelian version of the de Rham theorem — given essentially by the fundamental theorem of dg-algebraic rational homotopy theory — identifies this non-abelian real cohomology with coefficients in $\mathcal{A}$ with the non-abelian de Rham cohomology (2) with coefficients in the real-Whitehead $L_\infty$-algebra $\mathfrak{l}\mathcal{A}$ of $\mathcal{A}$.

For the case that $\mathcal{A} = KU_0$ the cohomology operation (4) coincides with the Chern character on complex topological K-theory, and generally for $\mathcal{A} = \mathcal{E}_n$ a term in an Omega-spectrum it coincides with the Chern-Dold character map on Whitehead-generalized cohomology (Prop. 7.2).

Therefore, it makes sense to refer to (4) generally as the character map on nonabelian cohomology taking values in non-abelian de Rham cohomology (FSS20, Part IV).

## References

Last revised on January 30, 2024 at 12:17:59. See the history of this page for a list of all contributions to it.