monoidal category

Noun

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