Monoidal Category - Examples

Examples

• Any category with finite products is monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example:
  • Set, the category of sets with the Cartesian product, one-element sets serving as the unit.
• Any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit.
• R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗_R serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit. As special cases one has:
  • K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit.
  • Ab, the category of abelian groups, with the group of integers Z serving as the unit.
• For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit.
• The category of pointed spaces is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as the unit.
• The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit.
• Bounded-above meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element.