An elementary proof of a criterion for subfunctors of Ext to be closed
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912406853124096 |
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| author | Cala, Juan Camilo |
| author_facet | Cala, Juan Camilo |
| contents | Let $\mathcal{A}$ be an abelian category and let $F$ be a subbifunctor of the additive bifunctor $\text{Ext}_{\mathcal{A}}^{1}(-,-)\colon \mathcal{A}^{\text{op}}\times \mathcal{A}\to \mathsf{Ab}$. Buan proved in [4] that $F$ is closed if, and only if, $F$ has the $3\times 3$-lemma property, a certain diagrammatic property satisfied by the class of $F$-exact sequences. The proof of this result relies on the theory of exact categories and on the Freyd--Mitchell embedding theorem, a very well-known overpowered result. In this paper we provide a proof of Buan's result only by means of elementary methods in abelian categories. To achieve this we survey the required theory of subfunctors leading us to a self-contained exposition of this topic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01203 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An elementary proof of a criterion for subfunctors of Ext to be closed Cala, Juan Camilo Category Theory Representation Theory 18E10, 18G15 (Primary) 18G99 (Secondary) Let $\mathcal{A}$ be an abelian category and let $F$ be a subbifunctor of the additive bifunctor $\text{Ext}_{\mathcal{A}}^{1}(-,-)\colon \mathcal{A}^{\text{op}}\times \mathcal{A}\to \mathsf{Ab}$. Buan proved in [4] that $F$ is closed if, and only if, $F$ has the $3\times 3$-lemma property, a certain diagrammatic property satisfied by the class of $F$-exact sequences. The proof of this result relies on the theory of exact categories and on the Freyd--Mitchell embedding theorem, a very well-known overpowered result. In this paper we provide a proof of Buan's result only by means of elementary methods in abelian categories. To achieve this we survey the required theory of subfunctors leading us to a self-contained exposition of this topic. |
| title | An elementary proof of a criterion for subfunctors of Ext to be closed |
| topic | Category Theory Representation Theory 18E10, 18G15 (Primary) 18G99 (Secondary) |
| url | https://arxiv.org/abs/2407.01203 |