Plethories and topology

A (formal) plethory is a rather complex algebraic structure; for the category-theory minded, it is a (ind-)representable endofunctor $P$ on the category of (graded) commutative $R$-algebras which also carries the structure of a comonoid with respect to composition: it has a comultiplication $P \to P \circ P$ which is coassociative and counital, but not cocommutative. Or to put it into explicit terms: it is an $R$-module $P$ with the following additional data:

  • a commutative multiplication $P \otimes_R P \to P$,
  • a cocommutative “coaddition” $P \to P \otimes P$,
  • a cocommutative “comultiplication” $P \to P \otimes P$, and
  • a “composition” $P \circ P \to P$, where it’s not entirely straightforward to describe what $\circ$ is.

Why would you care for this particular heap of structure? If you are a topologist, you might care because this is exactly the structure that (unstable) cohomology operations carry, for any cohomology theory $E$ such that $E_*E$ is flat over $E$. So not like stable homotopy.

We know a lot about unstable operations in singular cohomology, and we’ve gotten quite far without plethories. This is due to a fact that’s fairly particular to singular cohomology: the additive operations are a subset of the stable operations, and cohomology rings of Eilenberg-Mac Lane spaces are polynomial rings. These two facts together mean that we can describe all unstable operations as polynomials of a subset of the stable operations, the subset being defined by the excess condition. This is not true, for instance, for Morava $K$-theory.

I am currently trying to get my hands on the structure of the plethory of unstable operations in Morava $K$-theory, and see what I can say about

  • the $E_2$-term of the unstable Adams-Novikov spectral sequence, and unstable homotopy groups of $K(n)$-local spheres
  • The division functor (a la Lannes) in $K(n)$, injective and projective modules over the $K(n)$-plethory, and the $T$-functor
  • operations on $E_\infty$-spectra and their algebraic structure

I have a published article on the definition and structure of formal plethories and hopefully very soon a preprint on the structure of the $K(n)$-plethory via Dieudonné theory.