Commutativity and cocommutativity of cogroups in the category of connected graded algebras

Let ACGACG be the category of cogroups in the category AA of connected graded algebras over a fixed commutative ring R  . We study the full subcategory ACGco consisting of objects whose underlying algebras are graded commutative, together with the full subcategory AcoCGAcoCG consisting of cocommutative objects and the full subcategory ACGco consisting of objects whose underlying coalgebras are graded cocommutative. We establish categorical equivalences of these full subcategories with categories of simpler algebraic objects, and obtain the inclusion relation of the full subcategories. Since H⁎(ΩX;R)H⁎(ΩX;R) is a cogroup in AA for a 1-connected co-H-group (under the assumption that R is a field), the algebraic results are applied to the theory of co-H  -groups. We study when H⁎(ΩX;R)H⁎(ΩX;R) is in AcoCGAcoCG or ACGco, and generalize a theorem of Kachi.