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In category theory, a branch of mathematics, a symmetric monoidal category is a braided monoidal category that is maximally symmetric. That is, the braiding operator obeys an additional identity: . The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an space, so its group completion is an infinite loop space.〔R.W. Thomason, ("Symmetric Monoidal Categories Model all Connective Spectra" ), ''Theory and Applications of Categories'', Vol. 1, No. 5, 1995, pp. 78– 118.〕 ==Definition== A symmetric monoidal category is a monoidal category (''C'', ⊗) such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *:File:symmetric monoidal unit coherence.png *The associativity coherence: *:File:symmetric monoidal associativity coherence.png *The inverse law: *:File:symmetric monoidal inverse law.png In the diagrams above, ''a'', ''l'' , ''r'' are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「symmetric monoidal category」の詳細全文を読む スポンサード リンク
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