The functor that embeds the category of commutative algebras to associative algebras has the left adjoint - the quotient by the commutant ideal. For any dg-algebra $A$ let $A_{Ab}$ denote the derived functor of the quotient by the commutant ideal, i. e. take a free resolution of the algebra and then take the quotient by two-sided ideal generated by commutators. There is the canonical map $A \to A_{Ab}$.

Suppose that $A$ is the algebra of rational cohomology of a homotopy type $X$.

**My question is:** does $A_{Ab}$ has some geometric interpretation? More precisely:
does there exist a functor $M$ from homotopy types to homotopy types and a natural transformation $e: M \to id$ such that $H^*(M(X))=A_{Ab}$ and $e$ gives the canonical map?

There is a similar question on the geometric meaning of $A_{Ab}$ for $A=H_*(\Omega X)$, the homology of a loop space with the Pontryagin product. The answer to this question is simple: $A_{Ab}=H_*(\Omega^{\infty+1}\Sigma^{\infty} X)$ and the canonical map is induced by the canonical $\Omega X\to \Omega^{\infty+1}\Sigma^{\infty} X$.

Note that the Lie algebra of rational homotopy groups $\pi_{*-1}M(X)$ must be $L(U(\pi_{*-1} X))$, where $U$ is the universal enveloping algebra and $L$ makes an associative algebra into Lie algebra with the commutator as the Lie bracket.

It seems that my question is related with this one.