  Let A be a cogroup in the category of connected graded algebras over a
commutative ring R. Let nu denote the inverse of A and chi the antipode of the
underlying Hopf algebra of A. We clarify the differences and similarities of nu
and chi, and show that nu coincides with chi if and only if A is commutative as
a graded algebra. Let A^co_CG be the category of cogroups satisfying these
equivalent conditions. If R is a field, the category A^co_CG is completely
determined. We also establish an equivalence of the full subcategory of A^co_CG
consisting of objects of finite type with a full subcategory of the category of
positively graded R-modules without any assumption on R. The results in the
case of R=Q are applied to the theory of co-H-groups.
