In a context of Lie theory the Chevalley-Eilenberg algebra $CE(A)$ of an ∞-Lie groupoid $A$ is the algebra of functions on the $\infty$-Lie groupoid: a cosimplicial algebra which in degree $k$ is the algebra of functions on the k-morphisms of the $\infty$-Lie groupoid.
If $A = \mathfrak{g}$ is the infinitesimal object given by the Lie algebra of an ordinary Lie group, then $CE(A) = CE(\mathfrak{g})$ coinices (under taking its normalized chains) with the ordinary notion of Chevalley-Eilenberg algebra of a Lie algebra (with values). More generally, for $\mathfrak{a}$ an ∞-Lie algebroid, $CE(\mathfrak{a})$ is the corresponding Chevalley-Eilenberg dg-algebra.
The more general context in which the operation of forming Chevalley-Eilenberg algebras is to be understood is discussed at rational homotopy theory in an (∞,1)-topos.
Let $C = CartSp_{th}$, the category of infinitesimally thickened Cartesian spaces (see path ∞-groupoid for this notation).
There is an sSet-enriched Quillen adjunction
where $cAlg(\mathbb{R})^{op}$ is the opposite of the category of cosimplicial algebras over $\mathbb{R}$ equipped with the standard model structure on cosimplicial algebras.
The left adjoint $CE$ is the functor
where on the right we take degreewise homs out of the simplicial object $X \in sPSh(C) = [\Delta^{op},PSh(C)]$ into the presheaf $R$ represented by $\mathbb{R}$ and regard the result as an algebra by using the algebra structure on $\mathbb{R}$.
First notice that $CE$ is indeed an sSet-enriched functor: for $X,Y \in sPSh(C)$, an $n$-cell in the hom-complex $sPSh(X,Y)$ is a morphism $X \cdot \Delta[n] \to Y$. Applying $PSh_C(-,R)$ to that yields a morphism of cosimplicial algebras $PSh(Y,R) \to PSh(X,R) \otimes C^\bullet(\Delta[n])$. This is indeed an $n$-cell in $cAlg(PSh(Y,R), PSh(X,R))$ and the construction is evnidently compatible with composition on both sides
That $CE$ has a right adjoint
follows from general reasoning. In end/coend calculus we check
The $sSet$-enrichment of the right adjoint is then fixed by adjunction.
To see that $CE$ is a left Quillen functor, first observe that a cofibration $X \to Y$ in $sPSh(C)_{proj}^{loc}$ is the same as a cofibration in $sPSh(C)_{proj}$ which is in particular a cofibration in $sPSh(C)_{inj}$, which is a degreewise monomorphism. It follows that $Hom_{PSh(C)}(Y_n,R) \to Hom_{PSh(C)}(X_n,R)$ is a surjection for all $n \in \mathbb{N}$. Hence $CE$ preserves cofibrations.
Now observe that $CE$ send Cech nerve projections $C(\{U_i\}) \to U$ for covering families $\{U_i \to U\}$ to weak equivalences (A proof of this is spelled out for instance in section 8 here ). By the nature of Bousfield localization this implies that the right adjoint of $CE$ sends fibrant objects to locally fibrant objects.
Since $cAlg(\mathbb{R})^{op}$ is evidently left proper, since all its objects are cofibrant, this allows to apply HTT, prop. A.3.7.2, to conclude that $CE$ is a left Quillen functor.
Let $\mathfrak{g} := Lie(\mathbf{B}G) \hookrightarrow \mathbf{B}G$ be the ∞-Lie algebroid corresponding to $\mathbf{B}G$, which as a simplicial object in $\mathcal{T}$ is
where in each term we have the first order infinitesimal neighbourhood of the neutral element. Then $CE(\mathfrak{g})$ is (under passage to normalized cochains) the ordinary Chevalley-Eilenberg algebra of the Lie algebra of $G$.
This is discussed at Chevalley-Eilenberg algebra in synthetic differential geometry.
Let $X$ be an ordinary manifold. Using the notation and results from path ∞-groupoid we have that (under passage to normalized cochains) a canonical isomorphism
Then we have a zig-zag of quasi-isomorphism
in $cAlg(\mathbb{R})^{op}$. This is the de Rham theorem in that it exhibits the equivalence between de Rham cohomology and singular cohomology (details at path ∞-groupoid.)
Let more generally $X = (X_\bullet)$ be a simplicial manifold. Then
is isomorphic to the simplicial deRham complex on the right. See there for the proof.
The above proposition says that the simplicial deRham complexes of weakly equivalent simplicial manifolds are quasi-isomorphic.