In rational homotopy theory, one uses various algebras and coalgebras to model (simply connected) spaces (topological spaces or simplicial sets, usually) up to rational equivalences.
Two types of models one can use are dg cocommutative algebras (Quillen) and dg commutative algebras (Sullivan). I will leave the dg implicit from now on. The dual of a cocommutative coalgebra is always a commutative algebra (while the converse is only true in the finite dimensional case). I have the following question.
Let $X$ be a simply connected space, and suppose $C$ is a cocommutative rational model for $X$. Under what assumptions is its linear dual $C^\vee$ a commutative rational model for $X$?
The question looks very natural to me, and I would be really surprised if nobody had studied it before. Any kind of answer or reference would be great.