Symmetric monoidal category

Definition

• The unit coherence:
• :
• The associativity coherence:
• :
• The inverse law:
• :

Examples

• The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
• The category of groups. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
• More generally, any category with finite products, that is, a cartesian monoidal category, is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object is the unit object.
• The category of bimodules over a ring R is monoidal, but not necessarily symmetric. If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
• Given a field k and a group, the category of all k-linear representations of the group is a symmetric monoidal category. Here the standard tensor product of representations is used.
• The categories and of stereotype spaces over are symmetric monoidal, and moreover, is a closed symmetric monoidal category with the internal hom-functor.

  Properties

Specializations

Generalizations