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Let A be a cogroup in the category of connected graded algebras over acommutative ring R. Let nu denote the inverse of A and chi the antipode of theunderlying Hopf algebra of A. We clarify the differences and similarities of nuand chi, and show that nu coincides with chi if and only if A is commutative asa graded algebra. Let A^co_CG be the category of cogroups satisfying theseequivalent conditions. If R is a field, the category A^co_CG is completelydetermined. We also establish an equivalence of the full subcategory of A^co_CGconsisting of objects of finite type with a full subcategory of the category ofpositively graded R-modules without any assumption on R. The results in thecase of R=Q are applied to the theory of co-H-groups.

Cogroups in the category of connected graded algebras whose inverse and antipode coincide