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](http://arxiv.org/pdf/math/0407227.pdf), 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](http://books.google.de/books?id=s6NnkQs3JBMC&lpg=PP1&hl=de&pg=PA336#v=onepage&q&f=false) (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](http://arxiv.org/abs/math/0407227v1), 2004



Classification of plethories over $\mathbb{Q}$
The preprint <https://arxiv.org/abs/1701.01314> of Magnus Carlson, "Classification of plethories in characteristic zero" answers the question about plethories over $\mathbb{Q}$ in the affirmative.


