nLab non-abelian de Rham cohomology





Special and general types

Special notions


Extra structure



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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

[ω]=[ω]αΩ dR n1(X)ω=dα+ω [\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)[ω]=[ω]ω^Ω dR n1(X×[0,1]) clsd{ω=ω^| 0, ω=ω^| 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 L_\infty -algebra valued differential forms

Ω dR(X;𝔞) clsdHom dgAlg(CE(𝔞),Ω dR (X)) \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 L_\infty-algebra 𝔞\mathfrak{a}, which reduces to the ordinary case for 𝔞b n1\mathfrak{a} \equiv b^{n-1} \mathbb{R} the line Lie n n -algebra.

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


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

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



With (1) it follows that the ordinary de Rham cohomology in degree nn is equivalently non-abelian de Rham cohomology with coefficients in the line Lie n-algebra b n1𝔲(1)b^{n-1}\mathfrak{u}(1):

H dR(X;b n1𝔲(1))H dR n(X;b n1). H_{dR}\big( X ;\, b^{n-1}\mathfrak{u}(1) \big) \;\; \simeq \;\; H^n_{dR}\big( X ;\, b^{n-1}\mathbb{R} \big) \,.


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.

See at geometry of physics – flux quantization the section Total flux in Nonabelian de Rham cohomology.


Recipient of non-abelian character map

For 𝒜\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 𝒜\mathcal{A} as the nonabelian cohomology classified by 𝒜\mathcal{A} (the non-abelian cohomology in degree=1 with coefficients in the loop space \infty -group Ω𝒜\Omega \mathcal{A} ):

(3)H(X;𝒜)π 0Maps(X,𝒜)π 0Maps(X,BΩ𝒜)H 1(X;Ω𝒜). 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

𝒜K(n,A) \mathcal{A} \,\equiv\, K(n,A)

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

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

or if

𝒜KU 0BU× \mathcal{A} \;\equiv\; KU_0 \,\simeq\, B U \times \mathbb{Z}

is the classifying space KU0_0 for complex topological K-theory, then this reduces to to complex topological K-theory:

H(X;KU 0)K(X). H\big( X ;\, KU_0 \big) \;\; \simeq \;\; K(X) \,.

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

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

coincides with the Whitehead-generalized E E -cohomology.

Now the rationalization-unit η 𝒜 :𝒜𝒜 \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 𝒜\mathcal{A}

(4)H(;𝒜)H(;𝒜 )H(;𝒜 )H dR(;𝔩𝒜), 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 𝒜\mathcal{A} with the non-abelian de Rham cohomology (2) with coefficients in the real-Whitehead L L_\infty -algebra 𝔩𝒜\mathfrak{l}\mathcal{A} of 𝒜\mathcal{A}.

For the case that 𝒜=KU 0\mathcal{A} = KU_0 the cohomology operation (4) coincides with the Chern character on complex topological K-theory, and generally for 𝒜= n\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).


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