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 [x]\mathbb{Z}[x] of integer-coefficient polynomials in one variable is the free ring? on 11, we have an isomorphism RRing([x],R)R \stackrel \sim \to \mathsf{Ring}(\mathbb{Z}[x], R) which sends an element rRr \in R to evaluation at rr, p(x)p(r)p(x) \mapsto p(r). If we let RR also be [x]\mathbb{Z}[x], this gives exactly “substitution” or “composition” of polynomials. A plethory is a ring which carries a substitution structure. The ring Λ\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 Ring([x],)\mathsf{Ring}(\mathbb{Z}[x],-) is the identity functor on Ring\mathsf{Ring}, and Ring(Λ,)\mathsf{Ring}(\Lambda,-) sends every ring to its ring of Witt vectors.


A biring is a (commutative) ring object in Ring op\mathsf{Ring}^{op}. The category of birings is Ring(Ring op) op\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 LAdj(Ring,Ring)\mathsf{LAdj}(\mathsf{Ring},\mathsf{Ring}), the category of left adjoints RingRing\mathsf{Ring} \to \mathsf{Ring}, and also to RAdj(Ring,Ring) op\mathsf{RAdj}(\mathsf{Ring},\mathsf{Ring})^{op}, opposite of the category of right adjoints RingRing\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 [x]\mathbb{Z}[x]. A generators-and-relations description of \odot can be found in (TW70).

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

Plethory over a ring

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


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

  • D. Tall and G. Wraith, Representable functors and operations on rings, Proc. London Math. Soc. 3 (1970), 619–643.

The term “plethory” was introduced here:

Last revised on June 14, 2021 at 05:44:48. See the history of this page for a list of all contributions to it.