The geometric meaning of the higher quotient by the commutant ideal 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.
 A: [I have replaced an earlier and less complete answer]
The question depends on the precise meaning of $A_{ab}$.  In the derived world, if we kill commutators then we create new potential commutators and so do not immediately end up with something commutative.  Killing commutators once is the same as taking Hochschild homology.  I think it is now known from work of Smith and McClure that when you apply HH to something with an action of the little $k$-cubes operad $C(k)$, you always get something with an action of $C(k+1)$ (ie it is "one step more commutative").  An associative algebra has an action of $C(1)$, so you can apply HH repeatedly and pass to a colimit to get something with an action of $C(\infty)$, which is an $E_\infty$ operad.  If we are working over $\mathbb{Q}$ then  $E_\infty$ algebras are essentially the same as commutative algebras.  I think that this procedure converts free associative algebras to free commutative algebras.
UPDATE: David Ben-Zvi is right that I have misremembered results that actually apply to Hochschild cohomology (rather than homology) here.  Nonetheless, what I said about repeatedly killing commutators makes some kind of intuitive sense, so I still wonder whether something along those lines could be true.
