2-Rig Extensions and the Splitting Principle

TL;DR

This paper generalizes the classical splitting principle to 2-rigs, categorifying the concept and proposing a conjecture about splitting objects via 2-rig maps, with proof in the case of Schur functors.

Contribution

It introduces a categorified splitting principle for 2-rigs and proves it for the free 2-rig on one object, expanding the theory of symmetric functions and affine categories.

Findings

Proves the conjecture for the free 2-rig on one object.

Develops the representation theory of affine categories.

Establishes a categorified splitting principle in the context of 2-rigs.

Abstract

Classically, the splitting principle says how to pull back a vector bundle in such a way that it splits into line bundles and the pullback map induces an injection on $K$-theory. Here we categorify the splitting principle and generalize it to the context of 2-rigs. A 2-rig is a kind of categorified "ring without negatives", such as a category of vector bundles with $\oplus$ as addition and $\otimes$ as multiplication. Technically, we define a 2-rig to be a Cauchy complete $k$-linear symmetric monoidal category where $k$ has characteristic zero. We conjecture that for any suitably finite-dimensional object $r$ of a 2-rig $\mathsf{R}$, there is a 2-rig map $E \colon \mathsf{R} \to \mathsf{R'}$ such that $E(r)$ splits as a direct sum of finitely many "subline objects" and $E$ has various good properties: it is faithful, conservative, essentially injective, and the induced map of Grothendieck rings $K(E) \colon K(\mathsf{R}) \to K(\mathsf{R'})$ is injective. We prove this conjecture for the free 2-rig on one object, namely the category of Schur functors, whose Grothendieck ring is the free $\lambda$-ring on one generator, also known as the ring of symmetric functions. We use this task as an excuse to develop the representation theory of affine categories - that is, categories enriched in affine schemes - using the theory of 2-rigs.