Streamline diffusion - Revision history

en>Yobot: Tagging using AWB (10703)

2015-01-07T12:13:59Z

Tagging using AWB (10703)

← Older revision		Revision as of 14:13, 7 January 2015
Line 1:		Line 1:
	~~In [[category theory]], a '''traced monoidal category'''~~ is ~~a category with some extra structure which gives a reasonable notion of feedback~~.		Hi there. My title is Sophia Meagher even though it is not the title on my birth certification. For a whilst I've been in Alaska but I will have to transfer in a year or two. To play domino is some thing I really appreciate doing. Distributing manufacturing has been his profession for some time.<br><br>Here is my page - psychic readings ([http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review Our Web Page])

	~~A '''traced symmetric monoidal category''' is~~ a ~~[[symmetric monoidal category]] '''C''' together with a family of functions~~
	~~:<math>\mathrm{Tr}^U_{X,Y}:\mathbf{C}(X\otimes U,Y\otimes U)\to\mathbf{C}(X,Y)</math>~~
	~~called a ''trace'~~'~~, satisfying the following conditions:~~
	* naturality in ~~''X'': for every <math>f:X\otimes U\~~to ~~Y\otimes U</math> and <math>g:X'\to X</math>,~~
	~~::<math>\mathrm{Tr}^U_{X,Y}(f)g=\mathrm{Tr}^U_{X',Y}(f(g\otimes U))</math>~~

	~~[[Image:Trace diagram naturality 1.svg\|thumb\|center\|400px\|Naturality~~ in ~~X]]~~

	* naturality in ''Y'': for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:Y\to Y'</math>,
	~~::<math>g\mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{X,Y'}((g\otimes U)f)</math>~~

	~~[[Image:Trace diagram naturality 2~~.~~svg\|thumb\|center\|400px\|Naturality in Y]]~~

	* dinaturality in ''U'': for every <math>f:X\otimes U\to Y\otimes U'</math> and <math>g:U'\to U</math>
	~~::<math>\mathrm{Tr}^U_{X,Y}((Y\otimes g)f)=\mathrm{Tr}^{U'}_{X,Y}(f(X\otimes g))</math>~~

	~~[[Image:Trace diagram dinaturality~~.~~svg\|thumb\|center\|400px\|Dinaturality in U]]~~

	* vanishing I: for ~~every <math>f:X\otimes I\to Y\otimes I</math>,~~
	~~::<math>\mathrm{Tr}^I_{X,Y}(f)=f</math>~~

	~~[[Image:Trace diagram vanishing~~.~~svg\|thumb\|center\|400px\|Vanishing I]]~~

	* vanishing II: for every <~~math~~>~~f:X\otimes U\otimes V\to Y\otimes U\otimes V~~<~~/math~~>
	~~::<math>\mathrm{Tr}^{U\otimes V}_{X,Y}~~(~~f)=\mathrm{Tr}^U_{X,Y}(\mathrm{Tr}^V_{X\otimes U,Y\otimes U}(f))</math>~~

	[[~~Image:Trace diagram associativity.svg\|thumb\|center\|400px\|Vanishing II]]~~

	* superposing: ~~for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:W\to Z<~~/~~math>,~~
	~~::<math>g\otimes \mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{W\otimes X,Z\otimes Y}(g\otimes f)<~~/~~math>~~

	~~[[Image:Trace diagram superposition~~.~~svg\|thumb\|center\|400px\|Superposing]]~~

	* yanking:
	~~::<math>\mathrm{Tr}^U_{U,U}(\gamma_{U,U})=U</math>~~
	~~(where <math>\gamma<~~/~~math> is the symmetry of the monoidal category).~~

	~~[[Image:Trace diagram yanking~~.~~svg\|thumb\|center\|400px\|Yanking]]~~

	== ~~Properties~~ ==
	* Every [[compact closed category]~~] admits a trace.~~

	* Given a traced monoidal category '''C''', the ''Int construction'' generates the free (in some bicategorical sense) ~~compact closure Int('''C''') of '''C'''.~~

	~~== References ==~~
	* {{cite journal
	~~\| author = [[André Joyal]], [[Ross Street]], [[Dominic Verity]]~~
	~~\| year = 1996~~
	~~\| title = Traced monoidal categories~~
	~~\| journal = Mathematical Proceedings of the Cambridge Philosophical Society~~
	~~\| volume = 3~~
	~~\| pages = 447–468~~
	~~\| doi = 10.1017/S0305004100074338~~
	}}

	~~[[Category:Monoidal categories]]~~


	~~{{categorytheory-stub}}~~

en>LennVator: /* Explanation */

2013-08-10T04:53:53Z

Explanation

← Older revision		Revision as of 06:53, 10 August 2013
Line 1:		Line 1:
	~~The writer~~ is ~~known as Wilber Pegues~~. ~~She~~ functions as a ~~journey agent but quickly she~~'~~ll be on her own~~. ~~I am really fond of handwriting but I can~~'~~t make it my profession truly~~. ~~Some time ago he selected~~ to ~~live in North Carolina~~ and ~~he doesn~~'~~t strategy on altering it~~.<br><br>~~my page~~ :: ~~psychic solutions by lynne~~ ([~~http~~://~~c045~~.~~danah~~.~~co.kr~~/~~home~~/~~index~~.~~php?document_srl~~=~~1356970&mid~~=~~qna c045~~.~~danah~~.co.kr])		In [[category theory]], a '''traced monoidal category''' is a category with some extra structure which gives a reasonable notion of feedback.

			A '''traced symmetric monoidal category''' is a [[symmetric monoidal category]] '''C''' together with a family of functions
			:<math>\mathrm{Tr}^U_{X,Y}:\mathbf{C}(X\otimes U,Y\otimes U)\to\mathbf{C}(X,Y)</math>
			called a ''trace'', satisfying the following conditions:
			* naturality in ''X'': for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:X'\to X</math>,
			::<math>\mathrm{Tr}^U_{X,Y}(f)g=\mathrm{Tr}^U_{X',Y}(f(g\otimes U))</math>

			[[Image:Trace diagram naturality 1.svg\|thumb\|center\|400px\|Naturality in X]]

			* naturality in ''Y'': for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:Y\to Y'</math>,
			::<math>g\mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{X,Y'}((g\otimes U)f)</math>

			[[Image:Trace diagram naturality 2.svg\|thumb\|center\|400px\|Naturality in Y]]

			* dinaturality in ''U'': for every <math>f:X\otimes U\to Y\otimes U'</math> and <math>g:U'\to U</math>
			::<math>\mathrm{Tr}^U_{X,Y}((Y\otimes g)f)=\mathrm{Tr}^{U'}_{X,Y}(f(X\otimes g))</math>

			[[Image:Trace diagram dinaturality.svg\|thumb\|center\|400px\|Dinaturality in U]]

			* vanishing I: for every <math>f:X\otimes I\to Y\otimes I</math>,
			::<math>\mathrm{Tr}^I_{X,Y}(f)=f</math>

			[[Image:Trace diagram vanishing.svg\|thumb\|center\|400px\|Vanishing I]]

			* vanishing II: for every <math>f:X\otimes U\otimes V\to Y\otimes U\otimes V</math>
			::<math>\mathrm{Tr}^{U\otimes V}_{X,Y}(f)=\mathrm{Tr}^U_{X,Y}(\mathrm{Tr}^V_{X\otimes U,Y\otimes U}(f))</math>

			[[Image:Trace diagram associativity.svg\|thumb\|center\|400px\|Vanishing II]]

			* superposing: for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:W\to Z</math>,
			::<math>g\otimes \mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{W\otimes X,Z\otimes Y}(g\otimes f)</math>

			[[Image:Trace diagram superposition.svg\|thumb\|center\|400px\|Superposing]]

			* yanking:
			::<math>\mathrm{Tr}^U_{U,U}(\gamma_{U,U})=U</math>
			(where <math>\gamma</math> is the symmetry of the monoidal category).

			[[Image:Trace diagram yanking.svg\|thumb\|center\|400px\|Yanking]]

			== Properties ==
			* Every [[compact closed category]] admits a trace.

			* Given a traced monoidal category '''C''', the ''Int construction'' generates the free (in some bicategorical sense) compact closure Int('''C''') of '''C'''.

			== References ==
			* {{cite journal
			\| author = [[André Joyal]], [[Ross Street]], [[Dominic Verity]]
			\| year = 1996
			\| title = Traced monoidal categories
			\| journal = Mathematical Proceedings of the Cambridge Philosophical Society
			\| volume = 3
			\| pages = 447–468
			\| doi = 10.1017/S0305004100074338
			}}

			[[Category:Monoidal categories]]


			{{categorytheory-stub}}

en>David Eppstein: stub sort

2012-08-29T04:53:54Z

stub sort

New page

The writer is known as Wilber Pegues. She functions as a journey agent but quickly she'll be on her own. I am really fond of handwriting but I can't make it my profession truly. Some time ago he selected to live in North Carolina and he doesn't strategy on altering it.<br><br>my page :: psychic solutions by lynne ([http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna c045.danah.co.kr])