$C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$ There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. 
My question:
Is there any analogous statement for $C_\infty$-algebras? Recently I heard that a $C_\infty$-structure on $C$ corresponds to a dg. structure on the cofree Lie coalgebra generetad by $sC$ but I can't find any reference for that or prove it myself. 
 A: As Qiaochu Yuan mentions in a comment, Koszul duality of operads is an answer. I think the standard reference now is Algebraic operads by Loday and Vallette. It's a rather vast subject, so I'll try to make a (quick) sketch of it – read the book for more details. There is also the book Operads in algebra, topology and physics by Markl, Shnider, and Stasheff.
For all this answer, I will consider things happening in dg-modules (chain complexes) over a field.
Koszul duality is something that takes a (quadratic) operad $\mathtt{P}$ and spits out a dual cooperad $\mathtt{P}^¡$ together with a so-called twisting morphism $\kappa : \mathtt{P}^¡ \to \mathtt{P}$. This twisting morphism induces a morphism from the cobar construction $\Omega \mathtt{P}^¡$ (an operad) to $\mathtt{P}$.
When the operad $\mathtt{P}$ satisfies a special property called "being Koszul", this morphism $\Omega \mathtt{P}^¡ \to \mathtt{P}$ is a quasi-isomorphism, i.e. it induces an isomorphism on homology. The operad $\Omega \mathtt{P}^¡$ is then often denoted $\mathtt{P}_\infty$.
It's sometimes a bit easier to work with operads rather than with cooperads, and it's possible to construct a Koszul dual operad $\mathtt{P}^!$ from $\mathtt{P}$. This is an involution: $(\mathtt{P}^!)^! = \mathtt{P}$. The famous trinity of operads $\mathtt{Ass}$ (associative algebras), $\mathtt{Com}$ (commutative algebras), and $\mathtt{Lie}$ (Lie algebras) are all Koszul, and their duals are $\mathtt{Ass}^! = \mathtt{Ass}$, $\mathtt{Com}^! = \mathtt{Lie}$, and $\mathtt{Lie}^! = \mathtt{Com}$.
Now, what's the link with $A_\infty$- and $C_\infty$-algebras? It turns out that an $A_\infty$-algebra is the same thing as an algebra over $\mathtt{Ass}_\infty = \Omega \mathtt{Ass}^{¡}$, and a $C_\infty$-algebra is the same thing as an algebra over $\mathtt{Com}_\infty = \Omega \mathtt{Com}^¡$. This is not really a coincidence: when $\mathtt{P}$ is a Koszul operad, the operad $\mathtt{P}_\infty$ enjoys very nice properties. If $A$ is a dg-module equipped with a $\mathtt{P}$-algebra structure and $B$ is a dg-module quasi-isomorphic to $A$, then $B$ cannot necessarily be equipped with a $\mathtt{P}$-algebra structure; but it can be equipped with a $\mathtt{P}_\infty$, such that the quasi-isomorphism respect this structure. Moreover, a quasi-isomorphism $X \xrightarrow{\sim} Y$ of $\mathtt{P}$-algebras can always be inverted $X \xleftarrow{\sim} Y$, but the inverse is really an $\infty$-quasi-isomorphism of $\mathtt{P}_\infty$-algebras. (I believe these properties are what initially motivated the definition of $A_\infty$- and $C_\infty$-algebas, even before Koszul duality of operads was discovered.)
And now, Koszul duality allows one to reformulate the definition of $A_\infty$- and $C_\infty$-algebras. Since $\mathtt{Ass}^! = \mathtt{Ass}$, by some general abstract nonsense, to give $X$ an algebra structure over $A_\infty = \mathtt{Ass}_\infty$ is exactly the same thing as giving a square zero coderivation on $T^c(\Sigma X)$, the cofree coassociative (conilpotent) coalgebra on the suspension of $X$. And similarly, since $\mathtt{Com}^! = \mathtt{Lie}$, to give $X$ an algebra structure over $C_\infty = \mathtt{Com}_\infty$ is exactly the same thing as giving a square zero coderivation on $L^c(\Sigma X)$, the cofree Lie coalgebra on the suspension of $X$.
(And as a bonus, since $\mathtt{Lie}^! = \mathtt{Com}$, an $L_\infty$ structure on $X$ is thus the same thing as a square zero coderivation on $S^c(\Sigma X)$, the cofree cocommutative coalgebra on the suspension of $X$.) 
