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

Abstract

Let A CG be the category of cogroups in the category A of connected graded algebras over a fixed commutative ring R. We study the full subcategory A co CG consisting of objects whose underlying algebras are graded commutative, together with the full subcategory A coCG consisting of cocommutative objects and the full subcategory co A CG 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) is a cogroup in A 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) is in A coCG or A co CG , and generalize a theorem of Kachi.

Rational Cup Product and Algebraic K0-Groups of Rings of Continuous Functions

“…(2) The notion of a C 0 -space and the dual notion of a W 0 -space have been discussed in [5]; it is shown that X is a C 0 -space if and only if the rationalization of ΣX is a homotopy-cocommutative co-H-space. See [6] for another characterizing property of C 0 -spaces. Definition 1.3.…”