Rank functions on $(d+2)$-angulated categories -- a functorial approach
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913976739168256 |
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| author | Nkansah, David |
| author_facet | Nkansah, David |
| contents | We introduce the notion of a rank function on a $(d+2)$-angulated category $\mathcal{C}$ which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for $d$ an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in $\mathcal{C}$ and rank functions defined on morphisms in $\mathcal{C}$. Inspired by work of Conde, Gorsky, Marks and Zvonareva, for $d$ an odd positive integer, we show there is a bijective correspondence between rank functions on $\operatorname{\mathsf{Proj}}A$ and additive functions on $\operatorname{\mathsf{mod}}(\operatorname{\mathsf{Proj}}A)$, where $\operatorname{\mathsf{Proj}}A$ is endowed with the Amiot-Lin $(d+2)$-angulated category structure. This allows us to show that every integral rank function on $\operatorname{\mathsf{Proj}}A$ can be decomposed into irreducible rank functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19042 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rank functions on $(d+2)$-angulated categories -- a functorial approach Nkansah, David Representation Theory Category Theory Rings and Algebras 18G99, 18A25 (primary), 18G80, 18E10, 18G25 (secondary) We introduce the notion of a rank function on a $(d+2)$-angulated category $\mathcal{C}$ which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for $d$ an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in $\mathcal{C}$ and rank functions defined on morphisms in $\mathcal{C}$. Inspired by work of Conde, Gorsky, Marks and Zvonareva, for $d$ an odd positive integer, we show there is a bijective correspondence between rank functions on $\operatorname{\mathsf{Proj}}A$ and additive functions on $\operatorname{\mathsf{mod}}(\operatorname{\mathsf{Proj}}A)$, where $\operatorname{\mathsf{Proj}}A$ is endowed with the Amiot-Lin $(d+2)$-angulated category structure. This allows us to show that every integral rank function on $\operatorname{\mathsf{Proj}}A$ can be decomposed into irreducible rank functions. |
| title | Rank functions on $(d+2)$-angulated categories -- a functorial approach |
| topic | Representation Theory Category Theory Rings and Algebras 18G99, 18A25 (primary), 18G80, 18E10, 18G25 (secondary) |
| url | https://arxiv.org/abs/2405.19042 |