By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see e.g. Félix–Oprea–Tanré, *Algebraic models in geometry*, Theorem 5.52 for this formulation).

Maybe this is a naive question, but if I take a rational CDGA model (in the sense of Sullivan) $A$ of a space $X$, then $BA$ becomes a Hopf algebra. Is it then true that 1/ $BA$ is a rational CDGA model for $\Omega X$, and (if so) 2/, that the coproduct of $BA$ represents the product of $\Omega X$ upon realization?

(I am aware of this previous question, but if I understand it correctly it doesn't really answer my question... I've also been reading Majewski's book *Rational homotopical models and uniqueness* but I wasn't able to directly apply his results to my question.)

afterpassing to cohomology... (I feel it's one of these situations where the answer is right before my eyes but I can't see it...) $\endgroup$ – Najib Idrissi Feb 5 '16 at 15:13