nLab 2-plethory







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

The property that polynomials in [x]\mathbb{N}[x] can be composed is a consequence of the fact that [x]\mathbb{N}[x] is the free rig in one generator, and hence represents the identity functor RigRig\mathsf{Rig}\to \mathsf{Rig}. Similarly, the category of Schur functors Schur\mathsf{Schur} can be seen as a 2-rig (in the sense below) representing the identity 2-functor 2Rig2Rig2\mathsf{Rig}\to 2\mathsf{Rig}. As a consequence, Schur\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”.


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

A birig is equivalently defined as:

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

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

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

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

A rig-plethory is equivalently defined as:

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

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

The categorified versions of the definitions above are as follows.

The category 2Rig2\mathsf{Rig} has:

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

A 2-birig is a 2-rig B\mathsf B and a lift of the 2-functor 2Rig(B,):2RigCat\mathsf{2Rig}(B,-):\mathsf{2Rig}\to \mathsf{Cat} to a 2-functor Φ B:2Rig2Rig\Phi_\mathsf{B}:\mathsf{2Rig}\to \mathsf{2Rig} such that UΦ B=2Rig(B,)\mathsf U \circ \Phi_\mathsf{B}=\mathsf{2Rig}(\mathsf{B},-). Again, any two representable 2-functors Φ B\Phi_\mathsf{B}, Φ B\Phi_\mathsf{B'} can be composed. Denote the resulting object by BB\mathsf{B}\odot \mathsf{B'}, so that Φ BΦ B=Φ BB\Phi_\mathsf{B} \circ \Phi_\mathsf{B'}=\Phi_{\mathsf{B'}\odot \mathsf{B}}.

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

A 2-plethory can be equivalently defined as

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

  • A 2-comonad Φ:2Rig2Rig\Phi:\mathsf{2Rig} \to \mathsf{2Rig} whose underlying 2-functor is a right 2-adjoint.

Example: Schur functors

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

In particular, the identity 1:2Rig2Rig1: \mathsf{2Rig} \to \mathsf{2Rig} is a 2-plethory with underlying 2-rig kS¯\overline{k\mathsf{S}}. The set isomorphism classes J(kS¯)J(\overline{k\mathsf{S}}) has a commutative monoid structure coming from the decategorifications of \oplus and \otimes. Then, J(kS¯)J(\overline{k\mathsf{S}}) is isomorphic to the monoid of positive symmetric functions Λ +\Lambda_+. The 2-plethory structure induces the well-known rig-plethory structure on Λ +\Lambda_+. In turn, the Grothendieck group of Schur\mathsf{Schur} is K(kS¯)= J(kS¯)K(\overline{k\mathsf{S}})=\mathbb{Z}\otimes_\mathbb{N} J(\overline{k\mathsf{S}}), and the rig-plethory structure on J(kS¯)J(\overline{k\mathsf{S}}) extends to a ring-plethory structure on K(kS¯)K(\overline{k\mathsf{S}}).


Created on November 8, 2022 at 16:23:09. See the history of this page for a list of all contributions to it.