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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.
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| ==Formal definition==
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| 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>,
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| *<math> (f\otimes g)^\dagger=f^\dagger\otimes g^\dagger:B\otimes D\rightarrow A\otimes C </math>;
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| *<math> \alpha^\dagger_{A,B,C}=\alpha^{-1}_{A,B,C}:(A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)</math>;
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| *<math> \rho^\dagger_A=\rho^{-1}_A:A \rightarrow A \otimes I</math>;
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| *<math> \lambda^\dagger_A=\lambda^{-1}_A: A \rightarrow I \otimes A</math> and
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| *<math> \sigma^\dagger_{A,B}=\sigma^{-1}_{A,B}:B \otimes A \rightarrow A \otimes B</math>.
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| Here, <math>\alpha,\lambda,\rho</math> and <math>\sigma</math> are the [[natural isomorphism]]s that form the [[symmetric monoidal category|symmetric monoidal structure]].
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| ==Examples==
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| The following [[category (mathematics)|categories]] are examples of dagger symmetric monoidal categories:
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| * 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.
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| * 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]].
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| 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.
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| ==See also== | |
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| * [[Strongly ribbon category]]
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| ==References==
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| {{Reflist}}
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| [[Category:Dagger categories]]
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| [[Category:Monoidal categories]]
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