2-Rig Extensions and the Splitting Principle
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914967151706112 |
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| author | Baez, John C. Moeller, Joe Trimble, Todd |
| author_facet | Baez, John C. Moeller, Joe Trimble, Todd |
| contents | 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 $λ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 2-Rig Extensions and the Splitting Principle Baez, John C. Moeller, Joe Trimble, Todd Category Theory 05E05, 13D15, 18F30, 18M05, 20G05 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 $λ$-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. |
| title | 2-Rig Extensions and the Splitting Principle |
| topic | Category Theory 05E05, 13D15, 18F30, 18M05, 20G05 |
| url | https://arxiv.org/abs/2410.05598 |