In this paper, we mainly focus on extension-closed subcategories of extriangulated categories. Let $ {\mathcal{X}} $ be an extension-closed subcategory. We show that if $ C $ is $ {\mathcal{X}} $-projective and there is a minimal right almost split deflation in $ {\mathcal{X}} $ ending by $ C $, then there is an $ {\mathfrak{s}} $-triangle ending by $ C $ which is very similar to an Auslander-Reiten triangle in $ {\mathcal{X}} $. We also show that if the extriangulated category admits a negative first extension $ {\mathbb{E}}^{-1} $, and $ {\mathcal{X}} $ is self-orthogonal with respect to $ {\mathbb{E}}^{-1} $, then $ {\mathcal{X}} $ has an exact structure.
Citation: Lingling Tan, Tiwei Zhao. Extension-closed subcategories in extriangulated categories[J]. AIMS Mathematics, 2022, 7(5): 8250-8262. doi: 10.3934/math.2022460
In this paper, we mainly focus on extension-closed subcategories of extriangulated categories. Let $ {\mathcal{X}} $ be an extension-closed subcategory. We show that if $ C $ is $ {\mathcal{X}} $-projective and there is a minimal right almost split deflation in $ {\mathcal{X}} $ ending by $ C $, then there is an $ {\mathfrak{s}} $-triangle ending by $ C $ which is very similar to an Auslander-Reiten triangle in $ {\mathcal{X}} $. We also show that if the extriangulated category admits a negative first extension $ {\mathbb{E}}^{-1} $, and $ {\mathcal{X}} $ is self-orthogonal with respect to $ {\mathbb{E}}^{-1} $, then $ {\mathcal{X}} $ has an exact structure.
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Lingling Tan, Tiwei Zhao. Extension-closed subcategories in extriangulated categories[J]. AIMS Mathematics, 2022, 7(5): 8250-8262. doi: 10.3934/math.2022460
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