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arXiv · 2509.08144

Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves

Jaiung Jun·Alex Sistko·Cameron Wright
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Abstract

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 Andr\'e. As an application, we establish that the category $\textbf{TRS}_\bullet^\Sigma$ of tropical toric reflexive sheaves associated to a fan $\Sigma$, 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.

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Jaiung Jun, Alex Sistko, Cameron Wright. 2025-09-09. Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves. https://arxiv.org/abs/2509.08144

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