Monoidal Categories

Monoidal Categories

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.

• It is called monoidal because the structure is “monoid-like”. Any monoid forms a small monoidal category with object set , as monoidal product and the identity of as its identity object.
• 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.
• The category of sets is a monoidal category. The monoidal product is the disjoint union of sets, and the monoidal unit is the empty set.

Braided Monoidal Category

A braided monoidal category is a monoidal category with an additional structure, a braiding, which is a natural isomorphism that is compatible with the associativity and unit constraints. A symmetric monoidal category is a braided monoidal category with the braiding satisfying the symmetry axiom: .

Monoidal Functor

A monoidal functor between monoidal categories and is a functor together with a morphism in and a natural transformation such that the following diagrams commute for all objects in :

Enriched Categories

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.