group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
and
nonabelian homological algebra
The de Rham theorem (named after Georges de Rham) asserts that the de Rham cohomology $H^n_{dR}(X)$ of a smooth manifold $X$ (without boundary) is isomorphic to the “ordinary” $\mathbb{R}$-valued cohomology, i.e. the singular or Čech cohomology with real coefficients $H^n(X, \mathbb{R})$.
The theorem has several dozens of different proofs. For example in the Čech approach one can make a double complex whose first row is the Čech complex of a covering and first column is the de Rham complex and other entries are mixed and use spectral sequence argument (see the textbook of Bott and Tu, or the geometry lectures book by Postnikov, semester III).
This is maybe best formulated, understood and proven in the context of abelian sheaf cohomology:
Write $\mathbb{R}_c$ for
the abelian group $\mathbb{R}$
regarded not as a Lie group with the standard manifold structure on $\mathbb{R}$ but as a topologically discrete group on the underlying set of $\mathbb{R}$
and then regarded as a sheaf on $X$: the constant sheaf that sends connected $U \subset X$ to the set of constant maps $U \to \mathbb{R}$.
Write $\mathbf{B}^n \mathbb{R}_c$ for the corresponding Eilenberg-MacLane object in chain complexes of sheaves of abelian groups: this is the complex of sheaves with $\mathbb{R}_c$ in degree $n$:
Next, write $\bar \mathbf{B}^n \mathbb{R}$ (without the subscript $c$!) for the Deligne complex for $\mathbb{R}$
(The notation here is borrowed from that used at motivation for sheaves, cohomology and higher stacks: we can think of $\bar \mathbf{B}^n \mathbb{R}$ as a differential refinement of the object $\mathbf{B}^n \mathbb{R}_c$).
Then we have:
“ordinary” $\mathbb{R}$-valued cohomology of $X$ is the abelian sheaf cohomology with coefficients in $\mathbf{B}^n \mathbb{R}_c$.
de Rham cohomology of $X$ is the abelian sheaf cohomology with coefficients in $\bar \mathbf{B}^n \mathbb{R}$ (this is semi-obvious, requires a bit more discussion).
the Poincare lemma says that every closed differential form is locally exact, and hence there is a quasi-isomorphism of chain complexes of sheaves
given by injecting for each $U \subset X$ the set $\mathbb{R}$ as the constant functions into $C^\infty(U,\mathbb{R})$.
It is this quasi-isomorphism of coefficient objects that induces the de Rham isomorphism of abelian sheaf cohomology groups, which is ordinarily written as
…
The equivalence on cohomology asserted by the de Rham theorem is but a decategorification of a more refined statement: a quasi-isomorphism of cochain complexes. This even respects the product structure:
for $X$ a smooth manifold there is an equivalence of A-infinity algebras
between the de Rham complex and the collection of singular cochains equipped with the cup product.
This is due to (Gugenheim, 1977).
The de Rham theorem also holds internally in the context of suitable smooth toposes $\mathcal{T}$ modelling the axioms of synthetic differential geometry.
Specifically
the internal singular chain complex in $\mathcal{T}$ is given as the $R$-linear dual of the free interal $R$-module on the internal hom objects $[\Delta^n,X]$, where $R$ is the internal incarnation of the real numbers;
the de Rham complex is givebn by differential forms in synthetic differential geometry.
The de Rham theorem in $\mathcal{T}$ then asserts that for $X$ a manifold regarded as an object in the well-adapted smooth topos $\mathcal{T}$ the morphism
in $\mathcal{T}$ is an isomorphism for all $p \in \mathbb{N}$. This implies the standard (external) de Rham theorem.
This is discussed in chapter IV of
A little bit a long these lines for diffeological spaces is also in
de Rham theorem
Standard textbook references include
Raoul Bott, Loring Tu, Algebraic topology and differential forms,
M M Postnikov, Lectures on geometry, vol. III, Differentiable manifolds
Arne Lorenz, Abstract de Rham theorem, pdf slides (exposition of the standard de Rham theorem)
In analytic geometry also
The refinement of the de Rham theorem from an isomorphism of cohomology groups to an equivalence of A-∞ algebras of cochains and forms was first stated in
proven using Chen’s iterated integrals.
A review is in section 3 of