Tall-Wraith Monoids

TL;DR

This paper explores the algebraic and categorical properties of Tall-Wraith monoids, extending their theoretical framework and providing concrete examples related to algebraic structures and cohomology theories.

Contribution

It introduces a categorical perspective on Tall-Wraith monoids, showing how free algebra functors produce such monoids and analyzing specific examples like finite rings.

Findings

Free V-algebra functor applied to monoids yields Tall-Wraith monoids.

Categorical framework for Tall-Wraith monoids developed.

Example of Tall-Wraith monoid from self maps of finite rings analyzed.

Abstract

Tall-Wraith monoids were introduced in MR2559638 ("The hunting of the Hopf ring") to describe the algebraic structure on the set of unstable operations of a suitable generalised cohomology theory. In this paper we begin the study of Tall-Wraith monoids in an algebraic and categorical setting. We show that for V a variety of algebras, applying the free V-algebra functor to a monoid in Set produces a Tall-Wraith monoid. We also study the example of the Tall-Wraith monoid defined by the self set-maps of a finite ring, an example closely related to the original motivation for Tall-Wraith monoids.