In a context of Lie theory the Chevalley-Eilenberg algebra of an ∞-Lie groupoid is the algebra of functions on the -Lie groupoid: a cosimplicial algebra which in degree is the algebra of functions on the k-morphisms of the -Lie groupoid.
If is the infinitesimal object given by the Lie algebra of an ordinary Lie group, then coinices (under taking its normalized chains) with the ordinary notion of Chevalley-Eilenberg algebra of a Lie algebra (with values). More generally, for an ∞-Lie algebroid, 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 , the category of infinitesimally thickened Cartesian spaces (see path ∞-groupoid for this notation).
The left adjoint is the functor
where on the right we take degreewise homs out of the simplicial object into the presheaf represented by and regard the result as an algebra by using the algebra structure on .
First notice that is indeed an sSet-enriched functor: for , an -cell in the hom-complex is a morphism . Applying to that yields a morphism of cosimplicial algebras . This is indeed an -cell in and the construction is evnidently compatible with composition on both sides
That has a right adjoint
The -enrichment of the right adjoint is then fixed by adjunction.
To see that is a left Quillen functor, first observe that a cofibration in is the same as a cofibration in which is in particular a cofibration in , which is a degreewise monomorphism. It follows that is a surjection for all . Hence preserves cofibrations.
Now observe that send Cech nerve projections for covering families 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 sends fibrant objects to locally fibrant objects.
where in each term we have the first order infinitesimal neighbourhood of the neutral element. Then is (under passage to normalized cochains) the ordinary Chevalley-Eilenberg algebra of the Lie algebra of .
This is discussed at Chevalley-Eilenberg algebra in synthetic differential geometry.
Then we have a zig-zag of quasi-isomorphism
in . 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 be a simplicial manifold. Then
is isomorphic to the simplicial deRham complex on the right. See there for the proof.