monoidal category

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monoidal category

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monoidal category (plural monoidal categories)

1. (category theory) A category ${\displaystyle {\mathcal {C}}}$ with a bifunctor ${\displaystyle \otimes :{\mathcal {C}}\times {\mathcal {C}}\rightarrow {\mathcal {C}}}$ which may be called tensor product, an associativity isomorphism ${\displaystyle \alpha _{A,B,C}:(A\otimes B)\otimes C\simeq A\otimes (B\otimes C)}$, an object ${\displaystyle I}$ which may be called tensor unit, a left unit natural isomorphism ${\displaystyle \lambda _{A}:I\otimes A\simeq A}$, a right unit natural isomorphism ${\displaystyle \rho _{A}:A\otimes I\simeq A}$, and some "coherence conditions" (pentagon and triangle commutative diagrams for those isomorphisms).