dagger symmetric monoidal category について

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dagger symmetric monoidal category ：ウィキペディア英語版

dagger symmetric monoidal category
A dagger symmetric monoidal category is a monoidal category

\langle\mathbb,\otimes, I\rangle

which also possesses a dagger structure; in other words, it means that this category comes equipped not only with a tensor in the category theoretic sense but also with dagger structure which is used to describe unitary morphism and self-adjoint morphisms in

\mathbb

that is, a form of abstract analogues of those found in FdHilb, the category of finite-dimensional Hilbert spaces. This type of category was introduced by Selinger〔P. Selinger, ''( Dagger compact closed categories and completely positive maps )'', Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30 - July 1, 2005.〕 as an intermediate structure between dagger categories and the 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

\mathbb

which also has a dagger structure such that for all

f:A\rightarrow B

g:C\rightarrow D

and all

A,B

and

C

Ob(\mathbb)

,
*

(f\otimes g)^\dagger=f^\dagger\otimes g^\dagger:B\otimes D\rightarrow A\otimes C

;
*

\alpha^\dagger_=\alpha^_:(A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)

;
*

\rho^\dagger_A=\rho^_A:A \rightarrow A \otimes I

;
*

\lambda^\dagger_A=\lambda^_A: A \rightarrow I \otimes A

and
*

\sigma^\dagger_=\sigma^_:B \otimes A \rightarrow A \otimes B

.
Here,

\alpha,\lambda,\rho

and

\sigma

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