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| In [[mathematics]], a '''dagger category''' (also called '''involutive category''' or '''category with involution''' <ref name="Burgin" /><ref name="Lambek" />) is a [[category (mathematics)|category]] equipped with a certain structure called ''dagger'' or ''involution''. The name dagger category was coined by Selinger.<ref name="Selinger" />
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| == Formal definition ==
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| A '''dagger category''' is a [[category (mathematics)|category]] <math>\mathbb{C}</math> equipped with an [[Involution (mathematics)|involutive]], identity-on-object [[functor]] <math>\dagger\colon \mathbb{C}^{op}\rightarrow\mathbb{C}</math>.
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| In detail, this means that it associates to every [[morphism]] <math>f\colon A\to B</math> in <math>\mathbb{C}</math> its [[adjugate matrix|adjoint]] <math>f^\dagger\colon B\to A</math> such that for all <math>f\colon A\to B</math> and <math>g\colon B\to C</math>,
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| * <math> \mathrm{id}_A=\mathrm{id}_A^\dagger\colon A\rightarrow A</math>
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| * <math> (g\circ f)^\dagger=f^\dagger\circ g^\dagger\colon C\rightarrow A</math>
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| * <math> f^{\dagger\dagger}=f\colon A\rightarrow B\,</math>
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| Note that in the previous definition, the term ''adjoint'' is used in the linear-algebraic sense, not in the category theoretic sense.
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| Some reputable sources <ref name="Springer" /> additionally require for a ''category with involution'' that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is ''a''<''b'' implies <math>a\circ c<b\circ c</math> for morphisms ''a'', ''b'', ''c'' whenever their sources and targets are compatible.
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| == Examples ==
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| * The [[category (mathematics)|category]] '''Rel''' of [[Category of relations|sets and relations]] possesses a dagger structure i.e. for a given [[Relation (mathematics)|relation]] <math> R:X\rightarrow Y </math> in '''Rel''', the relation <math>R^\dagger:Y\rightarrow X</math> is the [[inverse relation|relational converse]] of <math> R</math>.
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| *:A self-adjoint morphism is a [[symmetric relation]].
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| * The [[category (mathematics)|category]] '''FdHilb''' of [[Category of finite dimensional Hilbert spaces|finite dimensional Hilbert spaces]] also possesses a dagger structure: Given a [[linear map]] <math>f:A\rightarrow B</math>, the map <math>f^\dagger:B\rightarrow A</math> is just its [[Hermitian adjoint|adjoint]] in the usual sense.
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| == Remarkable morphisms ==
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| In a dagger category <math>\mathbb{C}</math>, a [[morphism]] <math> f</math> is called
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| * '''unitary''' if <math>f^\dagger=f^{-1}</math>;
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| * '''self-adjoint''' if <math> f=f^\dagger</math> (this is only possible for an [[endomorphism]] <math>f\colon A \to A</math>).
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| The terms ''unitary'' and ''self-adjoint'' in the previous definition are taken from the [[Category of finite dimensional Hilbert spaces|category of Hilbert spaces]] where the morphisms satisfying those properties are then [[Unitary transformation|unitary]] and [[self-adjoint]] in the usual sense.
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| == See also ==
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| {{Portal|Category theory}}
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| * [[Dagger symmetric monoidal category]]
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| * [[Dagger compact category]]
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| == References ==
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| <references>
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| <ref name="Selinger">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>
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| <ref name="Burgin">M. Burgin, ''Categories with involution and correspondences in γ-categories'', IX All-Union Algebraic Colloquium, Gomel (1968), pp.34–35; M. Burgin, ''Categories with involution and relations in γ-categories'', Transactions of the Moscow Mathematical Society, 1970, v. 22, pp. 161–228 </ref>
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| <ref name="Lambek">J. Lambek, ''Diagram chasing in ordered categories with involution'', Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307</ref>
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| <ref name="Springer">{{SpringerEOM| title=Category with involution | id=Category_with_involution | oldid=16991 | first=M.Sh. | last=Tsalenko }}</ref>
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| </references>
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| *{{nlab|id=dagger-category|title=Dagger category}}
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| [[Category:Dagger categories| ]]
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