For any (commutative) ring $k$, a $k$-plethory is a monoid object in the monoidal category of $k$-$k$-birings, that is, it is a biring $P$ equipped with an associative map of birings $\circ: P \otimes_k P \to P$ and unit $k \langle e \rangle \to P$.

In other words,

a $k$-plethory is a commutative k-algebra together with a comonad structure on the covariant functor it represents, much as a k-algebra is the same as a $k$-module that represents a comonad. So, just as a $k$-algebra is exactly the structure that knows how to act on a $k$-module, a $k$-plethory is the structure that knows how to act on a commutative $k$-algebra. (BB05)

The most famous example of such an object is $\Lambda$, the ring of symmetric polynomials in countably many variables, which is a $\mathbb{Z}$-plethory.

References

James Borger, Ben Wieland?, Plethystic algebra, Advances in Mathematics194 (2005), 246–283. (web)

Last revised on July 2, 2015 at 07:32:38.
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