Duplicial functors, descent categories and generalized Hopf modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910801034477568 |
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| author | Bartulović, Ivan Boiquaye, John Krähmer, Ulrich |
| author_facet | Bartulović, Ivan Boiquaye, John Krähmer, Ulrich |
| contents | Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^χ_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $χ$ between two comonads. For the key example associated to a bialgebra $H$, right $χ$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $χ$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duplicial functors, descent categories and generalized Hopf modules Bartulović, Ivan Boiquaye, John Krähmer, Ulrich Category Theory K-Theory and Homology 18C15, 19D55 Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^χ_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $χ$ between two comonads. For the key example associated to a bialgebra $H$, right $χ$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $χ$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples. |
| title | Duplicial functors, descent categories and generalized Hopf modules |
| topic | Category Theory K-Theory and Homology 18C15, 19D55 |
| url | https://arxiv.org/abs/2501.14561 |