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 $\text{𝖱}$, there is a 2-rig map $𝐸:\text{𝖱}\to \text{𝖱}\prime$ 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 $𝐾(𝐸):𝐾(\text{𝖱})\to 𝐾(\text{𝖱}\prime )$ 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 $𝜆$-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.

It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a “plethory”: a monoid in the category of birings with its substitution monoidal structure. We show that similarly the category of Schur functors is a “2-plethory”, which descends to give the plethory structure on symmetric functions. Thus, much of the structure of symmetric functions exists at a higher level in the category of Schur functors.