nLab P-ring

Let PP be a commutative ring together with two types of structure on the set-valued functor Hom(P,)Hom(P, -) it represents:

  1. a commutative ring structure, which means we can view it as an endofunctor on the category of commutative rings (so SpecPSpec P is a commutative ring scheme), and
  2. a comonad structure on this endofunctor.

Then a PP-ring is simply a commutative ring equipped with a co-action of this comonad.

This is similar to the situation for operators on abelian groups. Let AA be an abelian group; then Hom(A,)Hom(A, -) is already abelian-group-valued (since Ab is closed), so we don’t need to do step (1). Then a comonad structure on the endofunctor Hom(A,)Hom(A, -) is the same as a (possibly non-commutative) ring structure on AA.

The moral of the story is that operators on abelian groups naturally form rings, and operators on commutative rings naturally form PPs. What you call such a PP is another matter. Borger and Wieland called such a thing a plethory, with a wink at plethysm, which is another name for composition of symmetric functions. Other people take a historical approach and use ‘Tall–Wraith’ combined with some other words. Probably the most descriptive and generalizable name would be something like ‘composition object in the category of commutative rings’.

Alternatively, the general naming scheme ‘plethystic object? in the category CC’ could be used, where CC could be the category of commutative rings, the category of abelian groups, and so on.


Last revised on July 6, 2010 at 11:22:41. See the history of this page for a list of all contributions to it.