In his paper [Cohomology $C\_\infty$-algebra and rational homotopy type](http://arxiv.org/abs/0811.1655v1), 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](https://mathoverflow.net/questions/50238/poincare-duality-and-the-a-infty-structure-on-cohomology/50255#50255) [answers](https://mathoverflow.net/questions/36444/definition-of-an-e-infinity-algebra/36478#36478).)


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. For the proof of Theorem 9.1 we seem to need that the functor $\mathcal{A}\Gamma$ applied to the weak equivalences of $C\_\infty$-algebras $$\lbrace f\_i\rbrace\colon (H(A),\lbrace m\_i\rbrace )\to (A,\lbrace d, \mu, 0,\ldots\rbrace)$$ yields a weak equivalence $\mathcal{A}\Gamma(H(A))\to\mathcal{A}\Gamma(A)$ in $CDGAlg$, but this is not stated anywhere and is not obvious to me.



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> Is the above true, and can anyone explain why?
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> Where can I read more about the functors $\Gamma$ and $\mathcal{A}$ and their properties, in particular in the setting of $C\_\infty$-algebras?
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Reference for functors in Kadeishvili's C_\infty paper
You can find all the arguments in Chapter 11 of the book downloadable at <http://math.unice.fr/~brunov/Operads.html>. This chapter deals with the bar and the cobar constructions for algebras over a Koszul operad. The last section [11.4] treats the extension to homotopy algebras. 


The theorem "the bar-cobar construction for $C\_\infty$-algebras sends $\infty$-quasi-isomorphisms to quasi-isomorphisms" is exactly Proposition 11.4.11 apply to the operad $P=Com$.


[Needless to say that this reference does not provide the very first proof of this fact for $C\_\infty$-algebras. Let's just say that it is freely available on the net, so easily accessible.] 


