Homology theory valued in the category of bicommutative Hopf algebras

TL;DR

This paper develops homology theories valued in the category of bicommutative Hopf algebras, expanding the scope beyond traditional module-based theories and including new examples with diverse coefficients.

Contribution

It introduces methods to construct H-valued homology theories with novel coefficients, including extraordinary types, in the abelian category of bicommutative Hopf algebras.

Findings

Constructed H-valued homology theories with non-group and non-function Hopf algebra coefficients

Provided examples of both ordinary and extraordinary homology theories

Expanded the framework of generalized homology theories to bicommutative Hopf algebras

Abstract

The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category A. It is known that the category of bicommutative (i.e. commutative and cocommutative) Hopf algebras over a field k is an abelian category. Denote the category by H. In this paper, we give some ways to construct H-valued homology theories. As a main result, we give H-valued homology theories whose coefficients are neither group Hopf algebras nor function Hopf algebras. The examples contain not only ordinary homology theories but also extraordinary ones.