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.
e.g.
- The category of abelian groups
is a monoidal category with the usual tensor product of abelian groups and the group of integers as the monoidal unit.
Enriched Category
We call a category
a -category, or a category enriched in , where is a monoidal category, if
- For each pair of objects
in , there is an object in , called the hom object with domain and codomain . Every morphism in can be uniquely described by in . - For each triple of objects
in , there is a morphism in , called the composition; - For each object
in , there is a morphism in , called the identity of . And they make the associativity diagram
and the unity diagram
commutes for all objects
in .
Preadditive Category
A preadditive category is a category enriched over the monoidal category of abelian groups.