Strong monad: Difference between revisions
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A '''strong monad''' is a mathematical object defined using [[category theory]] that is used in theoretical [[computer science]]. In technical terms, a strong monad over a [[monoidal category]] (''C'', ⊗, I) is a [[monad (category theory)|monad]] (''T'', |
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:[[Image:Strong monad left unit.svg]], [[Image:Strong monad associative.svg]], |
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==External links== |
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* [https://ncatlab.org/nlab/show/strong+monad Strong monad] at the [[nLab]] |
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[[Category:Adjoint functors]] |
[[Category:Adjoint functors]] |
Latest revision as of 03:56, 28 July 2024
![]() | This article may be too technical for most readers to understand.(April 2022) |
A strong monad is a mathematical object defined using category theory that is used in theoretical computer science. In technical terms, a strong monad over a monoidal category (C, ⊗, I) is a monad (T,
commute for every object A, B and C (see Definition 3.2 in [1]).
If the monoidal category (C, ⊗, I) is closed then a strong monad is the same thing as a C-enriched monad.
Commutative strong monads
[edit]For every strong monad T on a symmetric monoidal category, a costrength natural transformation can be defined by
A strong monad T is said to be commutative when the diagram
commutes for all objects and .[2]
One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,
- a commutative strong monad defines a symmetric monoidal monad by
- and conversely a symmetric monoidal monad defines a commutative strong monad by
and the conversion between one and the other presentation is bijective.
References
[edit]- ^ Moggi, Eugenio (July 1991). "Notions of computation and monads" (PDF). Information and Computation. 93 (1): 55–92. doi:10.1016/0890-5401(91)90052-4.
- ^ Muscholl, Anca, ed. (2014). Foundations of software science and computation structures : 17th (Aufl. 2014 ed.). [S.l.]: Springer. pp. 426–440. ISBN 978-3-642-54829-1.
- Anders Kock (1972). "Strong functors and monoidal monads" (PDF). Archiv der Mathematik. 23: 113–120. doi:10.1007/BF01304852. S2CID 13246783.
- Jean Goubault-Larrecq, Slawomir Lasota and David Nowak (2005). "Logical Relations for Monadic Types". Mathematical Structures in Computer Science. 18 (6): 1169. arXiv:cs/0511006. doi:10.1017/S0960129508007172. S2CID 741758.
External links
[edit]- Strong monad at the nLab