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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Lugansk National Taras Shevchenko University
2016
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oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2016-07-12T10:09:40Z |
| collection |
OJS |
| language |
English |
| topic |
pairing of functors adjoint functors weak (co)monads Frobenius monads firm modules cofirm comodules separability 18A40 18C20 16T1 |
| 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 |
| topic_facet |
pairing of functors adjoint functors weak (co)monads Frobenius monads firm modules cofirm comodules separability 18A40 18C20 16T1 |
| format |
Article |
| author |
Wisbauer, Robert |
| author_facet |
Wisbauer, Robert |
| author_sort |
Wisbauer, Robert |
| title |
Weak Frobenius monads and Frobenius bimodules |
| title_short |
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_sort |
weak frobenius monads and frobenius bimodules |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2016 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/133 |
| work_keys_str_mv |
AT wisbauerrobert weakfrobeniusmonadsandfrobeniusbimodules |
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2025-07-17T10:30:56Z |
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2025-07-17T10:30:56Z |
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1837890131658276864 |