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 A, LR defines a...
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| Дата: | 2013 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2013
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.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. |
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irk-123456789-1522582019-06-10T01:25:47Z Regular pairings of functors and weak (co)monads Wisbauer, R. 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. 2013 Article 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. http://dspace.nbuv.gov.ua/handle/123456789/152258 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Wisbauer, R. |
| spellingShingle |
Wisbauer, R. Regular pairings of functors and weak (co)monads Algebra and Discrete Mathematics |
| author_facet |
Wisbauer, R. |
| author_sort |
Wisbauer, R. |
| title |
Regular pairings of functors and weak (co)monads |
| title_short |
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_sort |
regular pairings of functors and weak (co)monads |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/152258 |
| 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. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT wisbauerr regularpairingsoffunctorsandweakcomonads |
| first_indexed |
2023-05-20T17:37:53Z |
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2023-05-20T17:37:53Z |
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1796153731599826944 |