Dagger category - Revision history

en>Selinger: /* Examples */ Groupoid examples: noted what the unitaries are.

2014-10-19T22:25:29Z

Examples: Groupoid examples: noted what the unitaries are.

← Older revision		Revision as of 00:25, 20 October 2014
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173.228.123.253: /* Remarkable morphisms */

2014-03-03T05:58:05Z

Remarkable morphisms

← Older revision		Revision as of 07:58, 3 March 2014
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	~~A '''dagger symmetric monoidal category'''~~ is ~~a [[monoidal category]] <math>\langle\mathbb{C},\otimes,~~ I\rangle</math> which also possesses a [[dagger category\|dagger structure]]; in other words, it means that this category comes equipped not only with a [[monoidal category\|tensor]] in the [[category theory\|category theoretic]] sense but also with [[dagger category\|dagger structure]] which is ~~used to describe [[unitary operator\|unitary morphism]]~~ and ~~[[self-adjoint\|self-adjoint morphisms]] in <math>\mathbb{C}</math> that is, a form of abstract analogues of those found in~~ ''~~'FdHilb''',~~ the ~~[[category of finite dimensional Hilbert spaces]]. This type of [[category~~ (~~mathematics)\|category]] was introduced by Selinger<ref>P. Selinger, ''~~[http://~~www~~.~~mscs~~.~~dal.ca~~/~~~selinger/papers.html#dagger Dagger compact closed categories and completely positive maps~~]'~~', Proceedings of~~ the ~~3rd International Workshop on Quantum Programming Languages, Chicago, June 30 - July 1, 2005~~.~~</ref> as an intermediate structure between [[dagger category\|dagger categories]] and the [[dagger compact category\|dagger compact categories]] that are used~~ in ~~[[categorical quantum mechanics]], an area which now also considers dagger symmetric monoidal categories when dealing with infinite dimensional [[quantum mechanical]] concepts~~.		Alyson is what my spouse loves to call me but I don't like when people use my complete title. He is an order clerk and it's some thing he truly appreciate. My spouse doesn't like it the way I do but what I really like doing best psychics ([http://brazil.amor-amore.com/irboothe mouse click the next document]) is caving but I don't have the time recently. I've always loved living in Mississippi.<br><br>Feel free to visit my web-site - real [http://kjhkkb.net/xe/notice/374835 email psychic readings] ([http://chungmuroresidence.com/xe/reservation_branch2/152663 http://chungmuroresidence.com/xe/reservation_branch2/152663])

	~~==Formal definition==~~

	~~A '''dagger symmetric monoidal category''' is a [[symmetric monoidal category]]~~ <~~math~~>~~\mathbb{C}~~<~~/math~~> ~~which also has a~~ [~~[dagger category\|dagger structure]] such that for all <math>f~~:~~A\rightarrow B <~~/~~math>, <math>g:C\rightarrow D <~~/~~math> and all <math> A,B<~~/~~math> and <math> C<~~/~~math> in <math>Ob(\mathbb{C})<~~/~~math>,~~
	*<math> (~~f\otimes g)^\dagger=f^\dagger\otimes g^\dagger~~:~~B\otimes D\rightarrow A\otimes C <~~/~~math>;~~
	*<math> \alpha^\dagger_{A,B,C}=\alpha^{-1}_{A,B,C}:(A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)</~~math>;~~
	*<math> \rho^\dagger_A=\rho^{-1}_A:A \rightarrow A \otimes I</~~math>;~~
	*<math> \lambda^\dagger_A=\lambda^{-1}_A: ~~A \rightarrow I \otimes A<~~/~~math> and~~
	*<math> \sigma^\dagger_{A,B}=\sigma^{-1}_{A,B}:B \otimes A \rightarrow A \otimes B</~~math>~~.
	~~Here, <math>\alpha,\lambda,\rho<~~/~~math> and <math>\sigma<~~/~~math> are the [[natural isomorphism]]s that form the [[symmetric monoidal category\|symmetric monoidal structure]].~~

	~~==Examples==~~

	~~The following [[category (mathematics)\|categories]] are examples of dagger symmetric monoidal categories:~~

	* The [[category (mathematics)\|category]] '''Rel''' of [[Category of relations\|sets and relations]~~] where the tensor is given by the [[Product (category theory)\|product]] and where the dagger of a relation is given by its relational converse.~~
	* The [[category (mathematics)\|category]] '''FdHilb''' of [[Category of finite dimensional Hilbert spaces\|finite dimensional Hilbert spaces]] is a dagger symmetric monoidal category where the tensor is the usual [[tensor product]] of Hilbert spaces and where the dagger of a [[linear map]] is given by its [[hermitian adjoint]].

	~~A dagger-symmetric category which is also [[compact closed category\|compact closed]] is a [[dagger compact category]]; both of the above examples are in fact dagger compact.~~

	~~==See also==~~

	* [[Strongly ribbon category]]

	~~==References==~~

	~~{{Reflist}}~~

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

en>Multipundit: /* References */

2013-03-14T23:55:07Z

References

← Older revision		Revision as of 01:55, 15 March 2013
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	~~Hi there~~. ~~Allow me start~~ by ~~introducing the author~~, ~~her name is Sophia~~. ~~What me~~ and ~~my family love is bungee jumping but I~~'~~ve been using~~ on ~~new issues recently~~. ~~For a whilst I've been~~ in ~~Alaska but I will have to transfer in a year or two~~. ~~Credit authorising~~ is ~~how he makes money.~~<br><br>~~My blog post online psychic reading~~ (~~[http~~://~~isaworld~~.pe.~~kr/?document_srl~~=~~392088 try this site~~])		A '''dagger symmetric monoidal category''' is a [[monoidal category]] <math>\langle\mathbb{C},\otimes, I\rangle</math> which also possesses a [[dagger category\|dagger structure]]; in other words, it means that this category comes equipped not only with a [[monoidal category\|tensor]] in the [[category theory\|category theoretic]] sense but also with [[dagger category\|dagger structure]] which is used to describe [[unitary operator\|unitary morphism]] and [[self-adjoint\|self-adjoint morphisms]] in <math>\mathbb{C}</math> that is, a form of abstract analogues of those found in '''FdHilb''', the [[category of finite dimensional Hilbert spaces]]. This type of [[category (mathematics)\|category]] was introduced by Selinger<ref>P. Selinger, ''[http://www.mscs.dal.ca/~selinger/papers.html#dagger Dagger compact closed categories and completely positive maps]'', Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30 - July 1, 2005.</ref> as an intermediate structure between [[dagger category\|dagger categories]] and the [[dagger compact category\|dagger compact categories]] that are used in [[categorical quantum mechanics]], an area which now also considers dagger symmetric monoidal categories when dealing with infinite dimensional [[quantum mechanical]] concepts.

			==Formal definition==

			A '''dagger symmetric monoidal category''' is a [[symmetric monoidal category]] <math>\mathbb{C}</math> which also has a [[dagger category\|dagger structure]] such that for all <math>f:A\rightarrow B </math>, <math>g:C\rightarrow D </math> and all <math> A,B</math> and <math> C</math> in <math>Ob(\mathbb{C})</math>,
			*<math> (f\otimes g)^\dagger=f^\dagger\otimes g^\dagger:B\otimes D\rightarrow A\otimes C </math>;
			*<math> \alpha^\dagger_{A,B,C}=\alpha^{-1}_{A,B,C}:(A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)</math>;
			*<math> \rho^\dagger_A=\rho^{-1}_A:A \rightarrow A \otimes I</math>;
			*<math> \lambda^\dagger_A=\lambda^{-1}_A: A \rightarrow I \otimes A</math> and
			*<math> \sigma^\dagger_{A,B}=\sigma^{-1}_{A,B}:B \otimes A \rightarrow A \otimes B</math>.
			Here, <math>\alpha,\lambda,\rho</math> and <math>\sigma</math> are the [[natural isomorphism]]s that form the [[symmetric monoidal category\|symmetric monoidal structure]].

			==Examples==

			The following [[category (mathematics)\|categories]] are examples of dagger symmetric monoidal categories:

			* The [[category (mathematics)\|category]] '''Rel''' of [[Category of relations\|sets and relations]] where the tensor is given by the [[Product (category theory)\|product]] and where the dagger of a relation is given by its relational converse.
			* The [[category (mathematics)\|category]] '''FdHilb''' of [[Category of finite dimensional Hilbert spaces\|finite dimensional Hilbert spaces]] is a dagger symmetric monoidal category where the tensor is the usual [[tensor product]] of Hilbert spaces and where the dagger of a [[linear map]] is given by its [[hermitian adjoint]].

			A dagger-symmetric category which is also [[compact closed category\|compact closed]] is a [[dagger compact category]]; both of the above examples are in fact dagger compact.

			==See also==

			* [[Strongly ribbon category]]

			==References==

			{{Reflist}}

			[[Category:Dagger categories]]
			[[Category:Monoidal categories]]

en>Linas: /* References */ add nlab reference

2012-08-23T02:20:16Z

References: add nlab reference

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