Formal plethories

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.

💡 Research Summary

The paper introduces “formal plethories” as a new algebraic framework for handling unstable operations in generalized cohomology theories. Classical plethories are representable endofunctors on the category of commutative k‑algebras equipped with a comonoid structure under composition. However, for a homotopy‑commutative ring spectrum E, the cohomology of its representing spaces Eₙ typically fails to be flat over E_* and lacks a Künneth isomorphism, preventing E_*(Eₙ) from forming a genuine plethory.

To overcome this, the author works in the pro‑category of graded E_*‑algebras. For any CW‑complex X, the pro‑object \