Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912732774662144 |
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| author | Dayaram, Kishan Kumar Goswami, Amartya Janelidze, Zurab Rodelo, Diana Ferreira Van der Linden, Tim |
| author_facet | Dayaram, Kishan Kumar Goswami, Amartya Janelidze, Zurab Rodelo, Diana Ferreira Van der Linden, Tim |
| contents | Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we show how the context of a `noetherian form', recently introduced by the second and third authors, allows a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. This context covers a wide range of examples: module categories, the category of groups, of graded abelian groups, the categories of Lie algebras, of cocommutative Hopf algebras, the category of Heyting semilattices, of loops, the dual of the category of pointed sets, the category of modular/distributive lattices and modular connections, the category of sets and partial bijections, and many others. More generally, it includes all semi-abelian and Grandis exact categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_11769 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context Dayaram, Kishan Kumar Goswami, Amartya Janelidze, Zurab Rodelo, Diana Ferreira Van der Linden, Tim Category Theory 18G50, 20J05, 18E13, 20J15, 18D30, 08B05 Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we show how the context of a `noetherian form', recently introduced by the second and third authors, allows a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. This context covers a wide range of examples: module categories, the category of groups, of graded abelian groups, the categories of Lie algebras, of cocommutative Hopf algebras, the category of Heyting semilattices, of loops, the dual of the category of pointed sets, the category of modular/distributive lattices and modular connections, the category of sets and partial bijections, and many others. More generally, it includes all semi-abelian and Grandis exact categories. |
| title | Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context |
| topic | Category Theory 18G50, 20J05, 18E13, 20J15, 18D30, 08B05 |
| url | https://arxiv.org/abs/2303.11769 |