Regular pairings of functors and weak (co)monads
For functors \(L:\mathbb{A}\to \mathbb{B}\) and \(R:\mathbb{B}\to \mathbb{A}\) betweenany categories \(\mathbb{A}\) and \(\mathbb{B}\), a pairing is defined by maps, natural in \(A\in \mathbb{A}\) and \(B\in \mathbb{B}\),\[\xymatrix{{\rm Mor}_\mathbb{B} (L(A),B) \ar@<0.5ex>[r]^{\alph...
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/738 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543434601398272 |
|---|---|
| author | Wisbauer, Robert |
| author_facet | Wisbauer, Robert |
| author_sort | Wisbauer, Robert |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-26T00:47:27Z |
| description | For functors \(L:\mathbb{A}\to \mathbb{B}\) and \(R:\mathbb{B}\to \mathbb{A}\) betweenany categories \(\mathbb{A}\) and \(\mathbb{B}\), a pairing is defined by maps, natural in \(A\in \mathbb{A}\) and \(B\in \mathbb{B}\),\[\xymatrix{{\rm Mor}_\mathbb{B} (L(A),B) \ar@<0.5ex>[r]^{\alpha} & {\rm Mor}_\mathbb{A} (A,R(B))\ar@<0.5ex>[l]^{\beta}}.\] \((L,R)\) is an adjoint pair provided \(\alpha\) (or \(\beta\)) is a bijection. In this case the composition \(RL\) defines a monad on the category \(\mathbb{A}\), \(LR\) defines a comonad on the category \(\mathbb{B}\), and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors.For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors \((L,R,\alpha,\beta)\) with \(\alpha = \alpha\cdot \beta\cdot \alpha\) and \(\beta = \beta \cdot\alpha\cdot\beta\). Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on \(\mathbb{A}\) gives rise to a regular pairing between \(\mathbb{A}\) and the category of compatible (co)modules. |
| first_indexed | 2026-02-08T07:59:40Z |
| format | Article |
| id | admjournalluguniveduua-article-738 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:59:40Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-7382018-04-26T00:47:27Z Regular pairings of functors and weak (co)monads Wisbauer, Robert pairing of functors; adjoint functors; weak monads and comonads; \(r\)-unital monads; \(r\)-counital comonads 18A40, 18C20, 16T15 For functors \(L:\mathbb{A}\to \mathbb{B}\) and \(R:\mathbb{B}\to \mathbb{A}\) betweenany categories \(\mathbb{A}\) and \(\mathbb{B}\), a pairing is defined by maps, natural in \(A\in \mathbb{A}\) and \(B\in \mathbb{B}\),\[\xymatrix{{\rm Mor}_\mathbb{B} (L(A),B) \ar@<0.5ex>[r]^{\alpha} & {\rm Mor}_\mathbb{A} (A,R(B))\ar@<0.5ex>[l]^{\beta}}.\] \((L,R)\) is an adjoint pair provided \(\alpha\) (or \(\beta\)) is a bijection. In this case the composition \(RL\) defines a monad on the category \(\mathbb{A}\), \(LR\) defines a comonad on the category \(\mathbb{B}\), and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors.For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors \((L,R,\alpha,\beta)\) with \(\alpha = \alpha\cdot \beta\cdot \alpha\) and \(\beta = \beta \cdot\alpha\cdot\beta\). Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on \(\mathbb{A}\) gives rise to a regular pairing between \(\mathbb{A}\) and the category of compatible (co)modules. Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/738 Algebra and Discrete Mathematics; Vol 15, No 1 (2013) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/738/269 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | pairing of functors adjoint functors weak monads and comonads \(r\)-unital monads \(r\)-counital comonads 18A40 18C20 16T15 Wisbauer, Robert Regular pairings of functors and weak (co)monads |
| title | Regular pairings of functors and weak (co)monads |
| title_full | Regular pairings of functors and weak (co)monads |
| title_fullStr | Regular pairings of functors and weak (co)monads |
| title_full_unstemmed | Regular pairings of functors and weak (co)monads |
| title_short | Regular pairings of functors and weak (co)monads |
| title_sort | regular pairings of functors and weak (co)monads |
| topic | pairing of functors adjoint functors weak monads and comonads \(r\)-unital monads \(r\)-counital comonads 18A40 18C20 16T15 |
| topic_facet | pairing of functors adjoint functors weak monads and comonads \(r\)-unital monads \(r\)-counital comonads 18A40 18C20 16T15 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/738 |
| work_keys_str_mv | AT wisbauerrobert regularpairingsoffunctorsandweakcomonads |