monoidal category

English

Noun

monoidal category (plural monoidal categories)

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