nLab 2-plethory

Idea

A 2-plethory (Baez-Moeller-Trimble) is a categorification of a rig-plethory. Just as a rig-plethory is defined as a monoid in $(\mathsf{Birig}, \odot)$, a 2-plethory is a pseudomonoid in $(2\mathsf{Birig}, \odot)$.

The property that polynomials in $\mathbb{N}[x]$ can be composed is a consequence of the fact that $\mathbb{N}[x]$ is the free rig in one generator, and hence represents the identity functor $\mathsf{Rig}\to \mathsf{Rig}$. Similarly, the category of Schur functors $\mathsf{Schur}$ can be seen as a 2-rig (in the sense below) representing the identity 2-functor $2\mathsf{Rig}\to 2\mathsf{Rig}$. As a consequence, $\mathsf{Schur}$ has the structure of a 2-plethory. Taking isomorphism classes, one recovers the rig-plethory structure on the positive symmetric functions, justifying the term “categorified plethysm”.

Definitions

Let $k$ be a field of characteristic zero, and let $\mathsf{Rig}$ be the category of rigs and rig morphisms. Let $U:\mathsf{Rig} \to \mathsf{Set}$ be the forgetful functor.

A birig is equivalently defined as:

• A rig object in $\mathsf{Rig}^\text{op}$.

• A rig $B$ and a lift of the representable functor $\mathsf{Rig}(B,-):\mathsf{Rig}\to \mathsf{Set}$ to a functor $\Phi_B:\mathsf{Rig}\to \mathsf{Rig}$ such that $U \circ \Phi_B=\mathsf{Rig}(B,-)$. Observe that any two representable 2-functors $\Phi_B$, $\Phi_B'$ can be composed. Denote the resulting object by $B\odot B'$, so that $\Phi_B \circ \Phi_B'=\Phi_{B' \circ B}$.

• A functor $\Phi:\mathsf{Rig}\to \mathsf{Rig}$ that is a right adjoint.

Denote by $(\mathsf{Birig}, \odot)$ the monoidal category of birigs, viewed as a full subcategory of $\mathsf Rig$.

A rig-plethory is equivalently defined as:

• A monoid in $(\mathsf{Birig}, \odot)$.

• A birig $B$ such that $\Phi_B$ is a comonad.

The categorified versions of the definitions above are as follows.

The category $2\mathsf{Rig}$ has:

• Objects: symmetric monoidal $k$-linear categories for which the tensor product is bilinear on hom-spaces and which are Cauchy complete.

• 1-morphisms: symmetric monoidal linear functors.

• 2-morphisms: symmetric monoidal linear natural transformations.

Let $\mathsf{U}:\mathsf{2Rig} \to \mathsf{Cat}$ be the forgetful 2-functor.

A 2-birig is a 2-rig $\mathsf B$ and a lift of the 2-functor $\mathsf{2Rig}(B,-):\mathsf{2Rig}\to \mathsf{Cat}$ to a 2-functor $\Phi_\mathsf{B}:\mathsf{2Rig}\to \mathsf{2Rig}$ such that $\mathsf U \circ \Phi_\mathsf{B}=\mathsf{2Rig}(\mathsf{B},-)$. Again, any two representable 2-functors $\Phi_\mathsf{B}$, $\Phi_\mathsf{B'}$ can be composed. Denote the resulting object by $\mathsf{B}\odot \mathsf{B'}$, so that $\Phi_\mathsf{B} \circ \Phi_\mathsf{B'}=\Phi_{\mathsf{B'}\odot \mathsf{B}}$.

Denote by $(\mathsf{2Birig}, \odot)$ the monoidal 2-category of 2-birigs, viewed as a full sub-2-category of $\mathsf{2Rig}$.

A 2-plethory can be equivalently defined as

• A pseudomonoid in $(2\mathsf{Birig}, \odot)$.

• A 2-comonad $\Phi:\mathsf{2Rig} \to \mathsf{2Rig}$ whose underlying 2-functor is a right 2-adjoint.

Example: Schur functors

For any 2-plethory $\Phi$ with left 2-adjoint $\Psi$, the composition $\mathsf{U}\Phi$ is representable, with representing object $\Psi(\overline{k\mathsf{S}})$, where $\overline{k\mathsf{S}}$ is the Cauchy completion of the $k$-linearization of the permutation groupoid $\mathsf{S}$. In turn, the 2-rig $\overline{k\mathsf{S}}$ is equivalent to $\mathsf{Schur}$ made into a 2-rig with the pointwise tensor product.

In particular, the identity $1: \mathsf{2Rig} \to \mathsf{2Rig}$ is a 2-plethory with underlying 2-rig $\overline{k\mathsf{S}}$. The set isomorphism classes $J(\overline{k\mathsf{S}})$ has a commutative monoid structure coming from the decategorifications of $\oplus$ and $\otimes$. Then, $J(\overline{k\mathsf{S}})$ is isomorphic to the monoid of positive symmetric functions $\Lambda_+$. The 2-plethory structure induces the well-known rig-plethory structure on $\Lambda_+$. In turn, the Grothendieck group of $\mathsf{Schur}$ is $K(\overline{k\mathsf{S}})=\mathbb{Z}\otimes_\mathbb{N} J(\overline{k\mathsf{S}})$, and the rig-plethory structure on $J(\overline{k\mathsf{S}})$ extends to a ring-plethory structure on $K(\overline{k\mathsf{S}})$.

Related concepts

• biring

• plethory

• plethysm

• Schur functor

References

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

• 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