Formal plethories

TL;DR

This paper introduces formal plethories as a framework to study unstable operations in generalized cohomology theories, focusing on algebraic structures over Pr"ufer rings and their linear approximations.

Contribution

It develops a new framework for formal plethories, analyzing their algebraic structure and approximations in the context of Pr"ufer rings.

Findings

Formal plethories are colimits of representable functors and form comonoids.

Logarithmic functors provide linear approximations of formal plethories.

The framework applies to unstable operations in generalized cohomology theories.

Abstract

Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework to study the algebra of such functors, which I call formal plethories, in the case where $E_*$ is a Pr\"ufer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.