The snail lemma and the long homology sequence
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910038348529664 |
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| author | González, Julia Ramos Vitale, Enrico |
| author_facet | González, Julia Ramos Vitale, Enrico |
| contents | In the first part of the paper, we establish an homotopical version of the snail lemma (which is a generalization of the classical snake lemma). In the second part, we introduce the category $\mathbf{Seq}(\mathcal A)$ of sequentiable families of arrows in a category $\mathcal A$ and we compare it with the category of chain complexes in $\mathcal A.$ We apply the homotopy snail lemma to a morphism in $\mathbf{Seq}(\mathcal A)$ obtaining first a six-term exact sequence in $\mathbf{Seq}(\mathcal A)$ and then, unrolling the sequence in $\mathbf{Seq}(\mathcal A),$ a long exact sequence in $\mathcal A.$ When $\mathcal A$ is abelian, this sequence subsumes the usual long homology sequence obtained from an extension of chain complexes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The snail lemma and the long homology sequence González, Julia Ramos Vitale, Enrico Category Theory 18G50, 18G35, 18N40 In the first part of the paper, we establish an homotopical version of the snail lemma (which is a generalization of the classical snake lemma). In the second part, we introduce the category $\mathbf{Seq}(\mathcal A)$ of sequentiable families of arrows in a category $\mathcal A$ and we compare it with the category of chain complexes in $\mathcal A.$ We apply the homotopy snail lemma to a morphism in $\mathbf{Seq}(\mathcal A)$ obtaining first a six-term exact sequence in $\mathbf{Seq}(\mathcal A)$ and then, unrolling the sequence in $\mathbf{Seq}(\mathcal A),$ a long exact sequence in $\mathcal A.$ When $\mathcal A$ is abelian, this sequence subsumes the usual long homology sequence obtained from an extension of chain complexes. |
| title | The snail lemma and the long homology sequence |
| topic | Category Theory 18G50, 18G35, 18N40 |
| url | https://arxiv.org/abs/2503.06577 |