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| In [[category theory]], a '''strong monad''' over a [[monoidal category]] <math>({\mathcal C},\otimes,I)</math> is a [[monad (category theory)|monad]] <math>(T,\eta,\mu)</math> together with a [[natural transformation]] <math>t_{A,B} : A\otimes TB\to T(A\otimes B)</math>, called (''tensorial'') ''strength'', such that the diagrams
| | Irwin Butts is what my wife enjoys to contact me although I don't truly like being known as like that. Bookkeeping is her day occupation now. Body developing is one of the issues I love most. Years in the past we moved to North Dakota.<br><br>Review my site ... [http://www.videoworld.com/blog/211112 www.videoworld.com] |
| :[[Image:Strong monad left unit.png]], [[Image:Strong monad unit.png]],
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| <br>
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| :[[Image:Strong monad assoc.png]],
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| and
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| :[[Image:Strong monad mult.png]]
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| commute for every object <math>A</math>, <math>B</math> and <math>C</math>.
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| == Commutative strong monads ==
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| For every strong monad ''T'' on a [[symmetric monoidal category]], a ''costrength'' natural transformation can be defined by
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| :<math>t'_{A,B}=T(\gamma_{B,A})\circ t_{B,A}\circ\gamma_{TA,B} : TA\otimes B\to T(A\otimes B)</math>.
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| A strong monad ''T'' is said to be '''commutative''' when the diagram
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| :[[Image:Strong monad commutation.png]]
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| commutes for all objects <math>A</math> and <math>B</math>.
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| One interesting fact about commutative strong monads is that they are "the same as" [[lax monoidal functor|symmetric]] [[monoidal monad]]s. More explicitly,
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| * a commutative strong monad <math>(T,\eta,\mu,t)</math> defines a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> by
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| :<math>m_{A,B}=\mu_{A\otimes B}\circ Tt'_{A,B}\circ t_{TA,B}:TA\otimes TB\to T(A\otimes B)</math>
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| * and conversely a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> defines a commutative strong monad <math>(T,\eta,\mu,t)</math> by
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| :<math>t_{A,B}=m_{A,B}\circ(\eta_A\otimes 1_{TB}):A\otimes TB\to T(A\otimes B)</math>
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| and the conversion between one and the other presentation is bijective.
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| == References ==
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| *{{cite journal
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| | author = Anders Kock
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| | year = 1972
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| | title = Strong functors and monoidal monads
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| | journal = Archiv der Math.
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| | volume = 23
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| | pages = 113–120
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| | url = http://home.imf.au.dk/kock/SFMM.pdf
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| | doi = 10.1007/BF01304852
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| }}
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| *{{cite journal
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| | author = Eugenio Moggi
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| | year = 1991
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| | title = Notions of computation and monads
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| | journal = Information and Computation
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| | volume = 93
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| | issue = 1
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| | pages =
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| | url = http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf
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| }}
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| *{{cite journal
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| | author = Jean Goubault-Larrecq, Slawomir Lasota and David Nowak
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| | year = 2005
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| | title = Logical Relations for Monadic Types
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| | doi = 10.1017/S0960129508007172
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| | journal = Mathematical Structures in Computer Science
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| | volume = 18
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| | issue = 06
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| | pages = 1169
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| | arxiv = cs/0511006
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| }}
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| [[Category:Adjoint functors]]
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| [[Category:Monoidal categories]]
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Irwin Butts is what my wife enjoys to contact me although I don't truly like being known as like that. Bookkeeping is her day occupation now. Body developing is one of the issues I love most. Years in the past we moved to North Dakota.
Review my site ... www.videoworld.com