Monoidal Category
A monoidal category is a tuple
where
is a category; - functor
is called the monoidal product; is called the monoidal unit; is a natural isomorphism: for all -objects , called the associativity isomorphism; and are also natural isomorphisms that for all -objects , called the left unit isomorphism and the right unit isomorphism respectively. And the following axioms hold:
- Middle Unity Axiom:
- Pentagon Axiom:
Moreover, a strict monoidal category is a monoidal category in which the components of
, and are all identity morphisms.
Remark
The triangle diagram and the pentagon diagram are somewhat to define the behavior of the tensor product morphisms
etc.