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