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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.08144 |
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| _version_ | 1866909779820019712 |
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| author | Jun, Jaiung Sistko, Alex Wright, Cameron |
| author_facet | Jun, Jaiung Sistko, Alex Wright, Cameron |
| contents | We study the category $F$-$\textbf{Mat}_\bullet$ of matroids over an idyll $F$. We show that $F$-$\textbf{Mat}_\bullet$ is a proto-exact category, a non-additive generalization of an exact category by Dyckerhoff and Kapranov. We further show that $F$-$\textbf{Mat}_\bullet$ is proto-abelian in the sense of André. As an application, we establish that the category $\textbf{TRS}_\bullet^Σ$ of tropical toric reflexive sheaves associated to a fan $Σ$, introduced by Khan and Maclagan, is also proto-exact and proto-abelian. We then investigate the stability of modular tropical toric reflexive sheaves within the framework of proto-abelian categories and reformulate Harder-Narasimhan filtrations in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08144 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves Jun, Jaiung Sistko, Alex Wright, Cameron Category Theory Algebraic Geometry Combinatorics We study the category $F$-$\textbf{Mat}_\bullet$ of matroids over an idyll $F$. We show that $F$-$\textbf{Mat}_\bullet$ is a proto-exact category, a non-additive generalization of an exact category by Dyckerhoff and Kapranov. We further show that $F$-$\textbf{Mat}_\bullet$ is proto-abelian in the sense of André. As an application, we establish that the category $\textbf{TRS}_\bullet^Σ$ of tropical toric reflexive sheaves associated to a fan $Σ$, introduced by Khan and Maclagan, is also proto-exact and proto-abelian. We then investigate the stability of modular tropical toric reflexive sheaves within the framework of proto-abelian categories and reformulate Harder-Narasimhan filtrations in this setting. |
| title | Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves |
| topic | Category Theory Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2509.08144 |