Regular pairings of functors and weak (co)monads
For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B,
 MorB(L(A), B) ↔ MorA(A, R(B)).
 (L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
|---|---|
| Дата: | 2013 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2013
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/152258 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Regular pairings of functors and weak (co)monads / R. Wisbauer // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 127–154. — Бібліогр.: 23 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862726251636064256 |
|---|---|
| author | Wisbauer, R. |
| author_facet | Wisbauer, R. |
| citation_txt | Regular pairings of functors and weak (co)monads / R. Wisbauer // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 127–154. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B,
MorB(L(A), B) ↔ MorA(A, R(B)).
(L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category 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, α, β) with α = α ⋅ β ⋅ α and β = β ⋅ α ⋅ β. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on A gives rise to a regular pairing between A and the category of compatible (co)modules.
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| first_indexed | 2025-12-07T18:56:52Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-152258 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:56:52Z |
| publishDate | 2013 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Wisbauer, R. 2019-06-09T13:38:32Z 2019-06-09T13:38:32Z 2013 Regular pairings of functors and weak (co)monads / R. Wisbauer // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 127–154. — Бібліогр.: 23 назв. — англ. 1726-3255 2010 MSC:18A40, 18C20, 16T15. https://nasplib.isofts.kiev.ua/handle/123456789/152258 For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B,
 MorB(L(A), B) ↔ MorA(A, R(B)).
 (L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category 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, α, β) with α = α ⋅ β ⋅ α and β = β ⋅ α ⋅ β. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on A gives rise to a regular pairing between A and the category of compatible (co)modules. The author wants to thank Gabriella Böhm,Tomasz Brzeziński and Bachuki Mesablishvili for their interest in this work and for helpful comments on a previous version of this paper. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Regular pairings of functors and weak (co)monads Article published earlier |
| spellingShingle | Regular pairings of functors and weak (co)monads Wisbauer, R. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152258 |
| work_keys_str_mv | AT wisbauerr regularpairingsoffunctorsandweakcomonads |