CatDat

• Notation: $(R{-}\Mod,\otimes,R)$
• Underlying category: category of left modules over a ring
• Children: $(R{-}\Mod,\otimes,R)$, $(R{-}\Mod,\otimes,R)$
• Related symmetric monoidal categories: $(\Ab,\otimes,\IZ)$
• nLab Link

When $R$ is a commutative ring, we equip the category of left $R$-modules $R{-}\Mod$ with the usual symmetric monoidal structure, where $\otimes$ is the tensor product of modules and the monoidal unit is $R$. The associator is defined by $(a \otimes b) \otimes c \mapsto a \otimes (b \otimes c)$, the left unitor maps $1 \otimes a \mapsto a$, the right unitor maps $a \otimes 1 \mapsto a$, and the symmetry maps $a \otimes b \mapsto b \otimes a$.