nLab plethory

Idea

The concept plethory derives from that of plethysm (Borger-Wieland 05, p. 2), a certain form of composition in the theory of symmetric functions.

In what follows, all rings are assumed to be commutative and unital.

Since the ring $\mathbb{Z}[x]$ of integer-coefficient polynomials in one variable is the free ring? on $1$, we have an isomorphism $R \stackrel \sim \to \mathsf{Ring}(\mathbb{Z}[x], R)$ which sends an element $r \in R$ to evaluation at $r$, $p(x) \mapsto p(r)$. If we let $R$ also be $\mathbb{Z}[x]$, this gives exactly “substitution” or “composition” of polynomials. A plethory is a ring which carries a substitution structure. The ring $\Lambda$ of symmetric functions is another example.

The data of a plethory is also what is needed to represent a “natural operation” on the category of rings. For example $\mathsf{Ring}(\mathbb{Z}[x],-)$ is the identity functor on $\mathsf{Ring}$, and $\mathsf{Ring}(\Lambda,-)$ sends every ring to its ring of Witt vectors.

Definitions

A biring is a (commutative) ring object in $\mathsf{Ring}^{op}$. The category of birings is $\mathsf{Ring}(\mathsf{Ring}^{op})^{op}$. The extra op ensures that a biring homomorphism is a ring homomorphism, not a reversed ring homomorphism.

The category of birings is equivalent to $\mathsf{LAdj}(\mathsf{Ring},\mathsf{Ring})$, the category of left adjoints $\mathsf{Ring} \to \mathsf{Ring}$, and also to $\mathsf{RAdj}(\mathsf{Ring},\mathsf{Ring})^{op}$, opposite of the category of right adjoints $\mathsf{Ring} \to \mathsf{Ring}$. Under this equivalence, the category of birings inherits a monoidal structure induced from endofunctor composition. The monoidal product is called the substitution product, denoted by $\odot$. The unit object is the ring of polynomials $\mathbb{Z}[x]$. A generators-and-relations description of $\odot$ can be found in (TW70).

A plethory is a monoid in $(\mathsf{Biring}, \odot, \mathbb{Z}[x])$. Equivalently, a plethory is a right adjoint comonad $\mathsf{Ring} \to \mathsf{Ring}$. Equivalently, a plethory is a left adjoint monad $\mathsf{Ring} \to \mathsf{Ring}$.

Plethory over a ring

For any (commutative) ring $k$, a $k$-plethory is a monoid object in the monoidal category of $k$-$k$-birings, with respect to the composition product. That is, it is a biring $P$ equipped with an associative map of birings $\circ: P \odot_k P \to P$ 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. (BW05)

Related concepts

• biring

• Tall-Wraith monoid

• 2-plethory

References

The idea was introduced here, where it was called a “biring triple”:

• D. Tall, Gavin Wraith: Representable functors and operations on rings, Proc. London Math. Soc. s3-20 4, (1970) 619-643 [doi:10.1112/plms/s3-20.4.619, MR265348]

The term “plethory” was introduced here:

• James Borger, Ben Wieland, Plethystic algebra, Advances in Mathematics 194 (2005) 246–283 [web]

• John Baez, Joe Moeller, Todd Trimble: Schur functors and categorified plethysm [arXiv:2106.00190]