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
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
The de Rham complex (named after Georges de Rham) $\Omega^\bullet(X)$ of a space $X$ is the cochain complex that in degree $n$ has the differential forms (which may mean: Kähler differential forms) of degree $n$, and whose differential is the de Rham differential or exterior derivative.
As $X$ varies this constitutes an abelian sheaf of complexes.
The de Rham complex of a smooth manifold is the cochain complex which in degree $n \in \mathbb{N}$ has the vector space $\Omega^n(X)$ of degree-$n$ differential forms on $X$. The coboundary map is the deRham exterior derivative. The cohomology of the de Rham complex (hence the quotient of closed differential forms by exact differntial forms?) is de Rham cohomology. Under the wedge product, the deRham complex becomes a differential graded algebra. This may be regarded as the Chevalley–Eilenberg algebra of the tangent Lie algebroid $T X$ of $X$.
The corresponding abelian sheaf in this case defines a smooth spectrum via the stable Dold-Kan correspondence, see at smooth spectrum – Examples – De Rham spectra.
For smooth varieties $X$, algebraic de Rham cohomology is defined to be the hypercohomology of the de Rham complex $\Omega_X^\bullet$.
De Rham cohomology has a rather subtle generalization for possibly singular algebraic varieties due to (Grothendieck).
For analytic spaces
In the general context of cohesive homotopy theory in a cohesive (∞,1)-topos $\mathbf{H}$, for $A \in \mathbf{H}$ a cohesive homotopy type, then the homotopy fiber of the counit of the flat modality
may be interpreted as the de Rham complex with coefficients in $A$.
This is the codomain for the Maurer-Cartan form $\theta_{\Omega A}$ on $\Omega A$ in this generality. The shape of $\theta_{\Omega A}$ is the general Chern character on $\Pi(\Omega A)$.
For more on this see at
More precisely, $\flat_{dR} \Sigma A$ and $\Pi_{dR} \Omega A$ play the role of the non-negative degree and negative degree part, respectively of the de Rham complex with coefficients in $\Pi \flat_{dR} \Sigma A$. For more on this see at
See at Deligne complex
(…)
A useful introduction is
A classical reference on the algebraic version is
See also
Yves André, Comparison theorems between algebraic and analytic De Rham cohomology (pdf)
Mikhail Kapranov, DG-Modules and vanishing cycles (KapranovDGModuleVanishingCycle.pdf?)