Rational homotopy types of simply connected topological spaces have been classified by weak equivalence classes of commutative cochain algebras (Sullivan) and by isomorphism classes of minimal commutative A_∞-algebras (Kadeishvili). We classify rational homotopy types of the space X by using the (noncommutative) singular cochain complex C~*(X,Q), with additional structure given by the homotopies introduced by Baues, {E_(1,k)} and {F_(p,q)}. We show that if we modify the resulting B_∞-algebra structure on this algebra by requiring that its bar construction be a Hopf algebra up to a homotopy, then weak equivalence classes of such algebras classify rational homotopy types.
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