Weak Frobenius monads and Frobenius bimodules
As observed by Eilenberg and Moore (1965), for a monad \(F\) with right adjoint comonad \(G\) on any category \(\mathbb{A}\), the category of unital \(F\)-modules \(\mathbb{A}_F\) is isomorphic to the category of counital \(G\)-comodules \(\mathbb{A}^G\). The monad \(F\) is Frobenius provided we...
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| Date: | 2016 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2016
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/133 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543133705175040 |
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| author | Wisbauer, Robert |
| author_facet | Wisbauer, Robert |
| author_sort | Wisbauer, Robert |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2016-07-12T10:09:40Z |
| description | As observed by Eilenberg and Moore (1965), for a monad \(F\) with right adjoint comonad \(G\) on any category \(\mathbb{A}\), the category of unital \(F\)-modules \(\mathbb{A}_F\) is isomorphic to the category of counital \(G\)-comodules \(\mathbb{A}^G\). The monad \(F\) is Frobenius provided we have \(F=G\) and then \(\mathbb{A}_F\simeq \mathbb{A}^F\). Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between \(\mathbb{A}_F\) and the category of bimodules \(\mathbb{A}^F_F\) subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad \((F,m,\eta)\) and a weak comonad \((F,\delta,\varepsilon)\) satisfying \(Fm\cdot \delta F = \delta \cdot m = mF\cdot F\delta\) and \(m\cdot F\eta = F\varepsilon\cdot \delta\), the category of compatible \(F\)-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible \(F\)-comodules. |
| first_indexed | 2026-02-08T07:57:38Z |
| format | Article |
| id | admjournalluguniveduua-article-133 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:38Z |
| publishDate | 2016 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-1332016-07-12T10:09:40Z Weak Frobenius monads and Frobenius bimodules Wisbauer, Robert pairing of functors; adjoint functors; weak (co)monads; Frobenius monads; firm modules; cofirm comodules; separability 18A40, 18C20, 16T1 As observed by Eilenberg and Moore (1965), for a monad \(F\) with right adjoint comonad \(G\) on any category \(\mathbb{A}\), the category of unital \(F\)-modules \(\mathbb{A}_F\) is isomorphic to the category of counital \(G\)-comodules \(\mathbb{A}^G\). The monad \(F\) is Frobenius provided we have \(F=G\) and then \(\mathbb{A}_F\simeq \mathbb{A}^F\). Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between \(\mathbb{A}_F\) and the category of bimodules \(\mathbb{A}^F_F\) subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad \((F,m,\eta)\) and a weak comonad \((F,\delta,\varepsilon)\) satisfying \(Fm\cdot \delta F = \delta \cdot m = mF\cdot F\delta\) and \(m\cdot F\eta = F\varepsilon\cdot \delta\), the category of compatible \(F\)-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible \(F\)-comodules. Lugansk National Taras Shevchenko University 2016-07-12 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/133 Algebra and Discrete Mathematics; Vol 21, No 2 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/133/pdf Copyright (c) 2016 Algebra and Discrete Mathematics |
| spellingShingle | pairing of functors adjoint functors weak (co)monads Frobenius monads firm modules cofirm comodules separability 18A40 18C20 16T1 Wisbauer, Robert Weak Frobenius monads and Frobenius bimodules |
| title | Weak Frobenius monads and Frobenius bimodules |
| title_full | Weak Frobenius monads and Frobenius bimodules |
| title_fullStr | Weak Frobenius monads and Frobenius bimodules |
| title_full_unstemmed | Weak Frobenius monads and Frobenius bimodules |
| title_short | Weak Frobenius monads and Frobenius bimodules |
| title_sort | weak frobenius monads and frobenius bimodules |
| topic | pairing of functors adjoint functors weak (co)monads Frobenius monads firm modules cofirm comodules separability 18A40 18C20 16T1 |
| topic_facet | pairing of functors adjoint functors weak (co)monads Frobenius monads firm modules cofirm comodules separability 18A40 18C20 16T1 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/133 |
| work_keys_str_mv | AT wisbauerrobert weakfrobeniusmonadsandfrobeniusbimodules |