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Dagger category : ウィキペディア英語版
Dagger category
In mathematics, a dagger category (also called involutive category or category with involution 〔〔) is a category equipped with a certain structure called ''dagger'' or ''involution''. The name dagger category was coined by Selinger.〔
== Formal definition ==

A dagger category is a category \mathcal equipped with an involutive functor \dagger\colon \mathcal^\rightarrow\mathcal that is the identity on objects, where \mathcal^ is the opposite category.
In detail, this means that it associates to every morphism f\colon A\to B in \mathcal its adjoint f^\dagger\colon B\to A such that for all f\colon A\to B and g\colon B\to C,
* \mathrm_A=\mathrm_A^\dagger\colon A\rightarrow A
* (g\circ f)^\dagger=f^\dagger\circ g^\dagger\colon C\rightarrow A
* f^=f\colon A\rightarrow B\,
Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense.
Some sources 〔 define a category with involution to be a dagger category with the additional property 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 a\circ c for morphisms ''a'', ''b'', ''c'' whenever their sources and targets are compatible.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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