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

In other words,

a kk-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 kk-module that represents a comonad. So, just as a kk-algebra is exactly the structure that knows how to act on a kk-module, a kk-plethory is the structure that knows how to act on a commutative kk-algebra. (BB05)

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


  • James Borger, Ben Wieland?, Plethystic algebra, Advances in Mathematics 194 (2005), 246–283. (web)
Revised on July 2, 2015 07:32:38 by David Corfield (