We study the category F-Mat_* of matroids over an idyll F. We show that F-Mat_* is a proto-exact category, a non-additive generalization of an exact category by Dyckerhoff and Kapranov. We further show that F-Mat_* is proto-abelian in the sense of André. As an application, we establish that the category TRS_*^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.
Keywords: matroids, idylls, tropical toric vector bundles, tropical toric reflexive sheaves, proto-exact category, proto-abelian category, stability, Harder-Narasimhan filtration
2020 MSC: 18F99, 14T99, 05B99
Theory and Applications of Categories, Vol. 45, 2026, No. 28, pp 1147-1200.
Published 2026-05-14.
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