A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912340531740672 |
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| author | Cala, Juan C. Hernández, Shaira R. |
| author_facet | Cala, Juan C. Hernández, Shaira R. |
| contents | Given an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we introduce the $3 \times 3$-lemma property for subfunctors of $\mathbb{E}$ and prove that an additive subfunctor $\mathbb{F}$ of $\mathbb{E}$ is closed if, and only if, it satisfies this condition. This characterization extends a well known result by A. Buan (for abelian categories) to extriangulated categories. As an application of this result, we get a new equivalent condition to describe saturated proper classes $ξ$ in $\mathcal{C}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories Cala, Juan C. Hernández, Shaira R. Category Theory Representation Theory 18G80, 18G10, 18G25 Given an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we introduce the $3 \times 3$-lemma property for subfunctors of $\mathbb{E}$ and prove that an additive subfunctor $\mathbb{F}$ of $\mathbb{E}$ is closed if, and only if, it satisfies this condition. This characterization extends a well known result by A. Buan (for abelian categories) to extriangulated categories. As an application of this result, we get a new equivalent condition to describe saturated proper classes $ξ$ in $\mathcal{C}$. |
| title | A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories |
| topic | Category Theory Representation Theory 18G80, 18G10, 18G25 |
| url | https://arxiv.org/abs/2504.15579 |