Stable model semantics - Revision history

en>Frietjes at 15:35, 11 July 2014

2014-07-11T15:35:22Z

← Older revision		Revision as of 17:35, 11 July 2014
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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]],~~
	~~<br>~~
	~~:[[Image:Strong monad assoc.png]],~~
	~~and~~
	~~:[[Image:Strong monad mult.png]]~~
	~~commute for every object <math>A</math>, <math>B</math> and <math>C</math>.~~

	~~== Commutative strong monads ==~~

	~~For every strong monad ''T'' on a [[symmetric monoidal category]], a ''costrength'' natural transformation can be defined by~~
	~~:<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>.~~
	~~A strong monad ''T''~~ is ~~said to be '''commutative''' when~~ the ~~diagram~~
	~~:[[Image:Strong monad commutation~~.~~png]]~~
	~~commutes for all objects <math>A</math> and <math>B</math>.~~

	~~One interesting fact about commutative strong monads is that they are "~~the ~~same as" [[lax monoidal functor\|symmetric]] [[monoidal monad]]s~~. ~~More explicitly,~~
	* a commutative strong monad <~~math~~>~~(T,\eta,\mu,t)~~<~~/math> defines a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> by~~
	~~:<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>~~
	* and conversely a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> defines a commutative strong monad <math>(T,\eta,\mu,t)</math> by
	~~:<math~~>~~t_{A,B}=m_{A,B}\circ(\eta_A\otimes 1_{TB}):A\otimes TB\to T(A\otimes B)</math>~~
	~~and the conversion between one and the other presentation is bijective~~.

	~~== References ==~~
	*{{cite journal
	~~\| author = Anders Kock~~
	~~\| year = 1972~~
	~~\| title = Strong functors and monoidal monads~~
	~~\| journal = Archiv der Math~~.
	~~\| volume = 23~~
	~~\| pages = 113–120~~
	~~\| url = http://home~~.~~imf.au.dk/kock/SFMM.pdf~~
	~~\| doi = 10.1007/BF01304852~~
	}}
	*{{cite journal
	~~\| author = Eugenio Moggi~~
	~~\| year = 1991~~
	~~\| title = Notions of computation and monads~~
	~~\| journal = Information and Computation~~
	~~\| volume = 93~~
	~~\| issue = 1~~
	~~\| pages =~~
	~~\| url =~~ http://www.~~disi~~.~~unige.it~~/~~person~~/~~MoggiE/ftp/ic91~~.~~pdf~~
	}}
	*{{cite journal
	~~\| author = Jean Goubault-Larrecq, Slawomir Lasota and David Nowak~~
	~~\| year = 2005~~
	~~\| title = Logical Relations for Monadic Types~~
	~~\| doi = 10~~.~~1017/S0960129508007172~~
	~~\| journal = Mathematical Structures in Computer Science~~
	~~\| volume = 18~~
	~~\| issue = 06~~
	~~\| pages = 1169~~
	~~\| arxiv = cs/0511006~~
	}}

	~~[[Category:Adjoint functors]]~~
	~~[[Category:Monoidal categories]~~]

en>Dewritech: clean up, typo(s) fixed: i.e, → i.e., using AWB

2013-10-19T16:37:14Z

clean up, typo(s) fixed: i.e, → i.e., using AWB

@@ Line 1: / Line 1: @@
-== though Buy Mont Blanc Pen Australia ==
+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
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+== Commutative strong monads ==
-   <li>[http://www.dabblersconcord.com/member/364501 http://www.dabblersconcord.com/member/364501]</li>
+For every strong monad ''T'' on a [[symmetric monoidal category]], a ''costrength'' natural transformation can be defined by
+:<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>.
-   <li>[http://nbn3975.egloos.com/1074630/ http://nbn3975.egloos.com/1074630/]</li>
+A strong monad ''T'' is said to be '''commutative''' when the diagram
+:[[Image:Strong monad commutation.png]]
-   <li>[http://www.qe.hk/forum.php?mod=viewthread&tid=448653 http://www.qe.hk/forum.php?mod=viewthread&tid=448653]</li>
+commutes for all objects <math>A</math> and <math>B</math>.
-   <li>[http://avalontowncar.com/index.php?/member/956242/ http://avalontowncar.com/index.php?/member/956242/]</li>
+One interesting fact about commutative strong monads is that they are "the same as" [[lax monoidal functor|symmetric]] [[monoidal monad]]s. More explicitly,
+* a commutative strong monad <math>(T,\eta,\mu,t)</math> defines a symmetric monoidal monad <math>(T,\eta,\mu,m)</math> by
-  </ul>
+:<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>

en>Sun Creator: Typo fixing using AWB

2012-08-02T19:39:43Z

Typo fixing using AWB

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