In his paper [Cohomology $C_\infty$-algebra and rational homotopy type][1], Tornike Kadeishvili describes how the rational cohomology of a simply-connected space carries the structure of a $C_\infty$-algebra, and how this structure determines the rational homotopy type of the space. (The result has been mentioned before on MathOverflow, eg in [these][2] [answers][3].)

I'm trying to follow the proofs, which are somewhat light on details. In particular, they rely on an adjoint pair of functors
$$\Gamma\colon CDGAlg \rightleftarrows DGLieCoalg \colon \mathcal{A}$$ introduced in section 4.3, between the categories of commutative differential graded algebras, and differential graded Lie coalgebras. The functor $\Gamma$ is given as the composition $$\Gamma\colon CDGAlg\stackrel{B}{\to}DGBialg\stackrel{Q}{\to}DGLieCoalg$$ where $B$ is a bar construction and $Q$ is the functor of indecomposables. The adjoint functor $\mathcal{A}$ is dual to the Chevalley-Eilenberg functor. There is a standard weak equivalence $\mathcal{A}\Gamma(A)\to A$.

 I am struggling to find any reference to these functors in the papers cited in the bibliography. In particular, for the proofs to work it seems that $\mathcal{A}\Gamma$ must take weak equivalences of $C_\infty$-algebras to weak equivalences in $CDGAlg$, but this is not stated anywhere. Can anyone confirm this? Also,

>Where can I read more about the functors $\Gamma$ and $\mathcal{A}$ and their properties, in particular in the setting of $C_\infty$-algebras?


  [1]: http://arxiv.org/abs/0811.1655v1
  [2]: http://mathoverflow.net/questions/50238/poincare-duality-and-the-a-infty-structure-on-cohomology/50255#50255
  [3]: http://mathoverflow.net/questions/36444/definition-of-an-e-infinity-algebra/36478#36478