To every simplicial manifold is associated its simplicial deRham complex.
Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) functor from simplicial manifolds to dg-algebras, is homotopical?
For instance: simplicial manifolds naturally embed into the category of simplicial presheaves on the category of manifolds, on which we have the standard local model structure on simplicial presheaves. On dg-algebras there is the standard model structure on dg-algebras.
Is there any literature that discusses explicitly the respect of the simplicial deRham complex operation of the respective weak equivalences?