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 ⊕ as addition and ⊗ 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 R, there is a 2-rig map E: R → 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, and the induced map of Grothendieck rings K(E): K(R) → K(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 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.
Keywords: Grothendieck ring, lambda-ring, representation theory, splitting principle, 2-rig
2020 MSC: 05E05, 13D15, 18F30, 18M05, 20G05
Theory and Applications of Categories, Vol. 44, 2025, No. 31, pp 964-1019.
Published 2025-09-28.
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