# Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). The corresponding comonad on $\mathrm{CAlg}(k)$ is simply $\mathrm{Hom}_{\mathsf{Mod}(k)}(A,-)$. Are these all plethories in the case $k=\mathbb{Q}$? This was asked by Bergman and Hausknecht in their book (p. 336), after noticing that all the known interesting examples "split" for $\mathbb{Q}$-algebras. Perhaps meanwhile more is known?

I think an equivalent statement would be: Does every continuous endofunctor of $\mathrm{CAlg}(\mathbb{Q})$ factor through the forgetful functor $\mathrm{CAlg}(\mathbb{Q}) \to \mathsf{Mod}(\mathbb{Q})$?

References on plethories (in order of appearance):

• D. O. Tall, G. C. Wraith, Representable Functors and Operations on Rings, Proc. London Math. Soc. (1970) s3-20(4): 619-643
• G. M. Bergman, A. O. Hausknecht, Cogroups and co-rings in categories of associative rings, American Mathematical Society, Mathematical Surveys and Monographs # 45, 1996.
• J. Borger, B. Wieland, Plethystic algebra, arXiv, 2004
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As far as I know, this question is still open. The only result in this direction I'm aware of is the subject of a paper by Buium called "Arithmetic analogues of derivations" where he gives a complete classification of biring structures on the polynomial algebra in two variables. He doesn't even restrict to $\mathbf{Q}$ coefficients. The answer is that you only get the ones responsible for structures you knew about already: ring endomorphisms, derivations, Frobenius lifts/Witt vectors/$p$-derivations, and lifts of the mod $p$ identity map. (Over $\mathbf{Q}$-algebras, the last two agree with the first.) From what I remember, his argument only uses a little bit of the theory of commutative algebraic groups.
In my heart, I think plethories over $\mathbf{Q}$ and even $\mathbf{Z}$ should be classifiable. Basically you should be able to get anything by performing mod $p$ amplification constructions (as described in my paper with Wieland) to the ones coming from the linear ones. Someone should do this! It would be a classification theorem for the rest of time. I'd like to believe it'd be doable with some meditating on Buium's result and a bit of pushing.