Crossed Burnside rings for groupoids
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913089246461952 |
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| author | Shiizuka, Keitaro |
| author_facet | Shiizuka, Keitaro |
| contents | In this paper, we extend the classical theory of crossed $G$-sets and the crossed Burnside ring from a finite group $G$ to a finite groupoid $\mathcal{G}$. We introduce a natural monoidal structure on the category of crossed $\mathcal{G}$-sets over a $\mathcal{G}$-monoid and construct the corresponding crossed Burnside ring of a $\mathcal{G}$-monoid. Finally, we prove a decomposition theorem that expresses the crossed Burnside ring of a groupoid as a product of crossed Burnside rings of groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03401 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Crossed Burnside rings for groupoids Shiizuka, Keitaro Category Theory 19A22, 18B40, 20L05 In this paper, we extend the classical theory of crossed $G$-sets and the crossed Burnside ring from a finite group $G$ to a finite groupoid $\mathcal{G}$. We introduce a natural monoidal structure on the category of crossed $\mathcal{G}$-sets over a $\mathcal{G}$-monoid and construct the corresponding crossed Burnside ring of a $\mathcal{G}$-monoid. Finally, we prove a decomposition theorem that expresses the crossed Burnside ring of a groupoid as a product of crossed Burnside rings of groups. |
| title | Crossed Burnside rings for groupoids |
| topic | Category Theory 19A22, 18B40, 20L05 |
| url | https://arxiv.org/abs/2605.03401 |