Non-Abelian extensions of groupoids and their groupoid rings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866918259730677760 |
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| author | Machado, Natã Öinert, Johan Wagner, Stefan |
| author_facet | Machado, Natã Öinert, Johan Wagner, Stefan |
| contents | We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids $\mathcal{N} \to \mathcal{E} \to \mathcal{G}$ gives rise to a groupoid crossed product of $\mathcal{G}$ by the groupoid ring of $\mathcal{N}$ which recovers the groupoid ring of $\mathcal{E}$ up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03369 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-Abelian extensions of groupoids and their groupoid rings Machado, Natã Öinert, Johan Wagner, Stefan Rings and Algebras Category Theory Operator Algebras 20L05, 16S35, 16S99 (Primary) 16E40, 16W50 (Secondary) We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids $\mathcal{N} \to \mathcal{E} \to \mathcal{G}$ gives rise to a groupoid crossed product of $\mathcal{G}$ by the groupoid ring of $\mathcal{N}$ which recovers the groupoid ring of $\mathcal{E}$ up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products. |
| title | Non-Abelian extensions of groupoids and their groupoid rings |
| topic | Rings and Algebras Category Theory Operator Algebras 20L05, 16S35, 16S99 (Primary) 16E40, 16W50 (Secondary) |
| url | https://arxiv.org/abs/2207.03369 |