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In his paper Cohomology $C_\infty$-algebra and rational homotopy type, 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 answers.)

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.

Is the above true, and can anyone explain why?

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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This is standard Koszul duality. Take your favourite book on rational homotopy theory. –  Fernando Muro May 25 '11 at 10:10
    
Dear Fernando, please could you expand on your comment a little? I have two favourite books on RHT, and neither of them have Koszul duality in the index. –  Mark Grant May 25 '11 at 14:42
    
See for instance IV.22 in MR1802847 (2002d:55014) Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude Rational homotopy theory. Graduate Texts in Mathematics, 205. Springer-Verlag, New York, 2001. xxxiv+535 pp. ISBN: 0-387-95068-0 (Reviewer: John F. Oprea), 55P62 (18Gxx 55U35) –  Fernando Muro May 25 '11 at 15:40
    
@Fernando: Thanks. Do you mean that the functors in that chapter are be (linearly) dual to the ones described above? What is still worrying me is that the weak equivalence $H(A)\to A$ is not a CDGA map, in particular it is only multiplicative up to boundaries. –  Mark Grant May 26 '11 at 20:50
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2 Answers 2

up vote 4 down vote accepted

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.]

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Thanks Bruno. This certainly seems to be the right general framework to prove this, and a lot more besides! I've begun reading. –  Mark Grant May 31 '11 at 19:03
    
Feel free to ask if you have any further question. I imagine that you were looking for a more ad hoc reference. Read this one with $P=Com$ and $P^{anti !}=Lie^*$ in mind. –  Bruno V. Jun 1 '11 at 12:11
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Ben Walter and I make the functors $\Gamma$ and $A$ more explicit, by using an explicit model for the cofree Lie Coalgebra functor, in this paper. We do not discuss the application to $\infty$-algebras as Bruno does in much greater generality. (We were interested in using explicit models to be able to compute, in particular in the long exact sequence of a fibration as we do in a sequel to this paper on Hopf invariants.)

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Thanks Dev. The paper looks interesting, I'll take a look. –  Mark Grant Jun 1 '11 at 10:44
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