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Weak Frobenius monads and Frobenius bimodules

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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Main Author: Wisbauer, Robert
Format: Article
Language:English
Published: Lugansk National Taras Shevchenko University 2016
Subjects:
pairing of functors; adjoint functors; weak (co)monads; Frobenius monads; firm modules; cofirm comodules; separability
18A40
18C20
16T1
Online Access:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/133
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spelling 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
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
first_indexed 2024-04-12T06:26:03Z
last_indexed 2024-04-12T06:26:03Z
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