The Hunting of the Hopf Ring

TL;DR

This paper introduces a novel algebraic framework using Tall-Wraith monoids to describe unstable cohomology operations and their Hopf rings, enhancing understanding of their structure in generalized cohomology theories.

Contribution

It provides a new algebraic description of unstable cohomology operations as a graded, completed Tall-Wraith monoid, and explores modules and examples related to these structures.

Findings

Unstable cohomology operations form a graded, completed Tall-Wraith monoid.

The E^*-cohomology of a space is a module over this monoid.

The Hopf ring of unstable co-operations is also a module for the Tall-Wraith monoid.

Abstract

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.