group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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 $\omega \in \Omega_{dR}^n(X)_{clsd}$
is equivalent to the concordance-relation [FSS20, Prop. 6.4]
But the latter concordance-relation immediately generalizes to flat $L_\infty$-algebra valued differential forms
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]:
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 $X$:
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 $b^{n-1}\mathfrak{u}(1)$:
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.
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 $\Omega \mathcal{A}$ ):
For example, in the case that
is an Eilenberg-MacLane space for a discrete abelian group $A$, this reduces to ordinary cohomology:
or if
is the classifying space KU$_0$ for complex topological K-theory, then this reduces to to complex topological K-theory:
Generally, if $\mathcal{E}$ is an Omega-spectrum of spaces, then
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 $\mathcal{A}$
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).
Last revised on January 30, 2024 at 12:17:59. See the history of this page for a list of all contributions to it.