Category theory MOC

Monoidal category

A monoidal category is the vertical Categorification of a monoid. cat Explicitly, a monoidal category is equipped with¹

1. a functor called the tensor product;
2. an object called the tensor unit;
3. a natural isomorphism with components in [[Functor category|]] called the associator;
4. a natural isomorphism with components in [[Endofunctor category|]] called the left-unitor; and
5. a natural isomorphism with components in [[Endofunctor category|]] called the right-unitor;

satisfying the so-called triangle identity

and pentagon identity

Together these diagrams ensure that the operation of is unital associative up to natural isomorphism, by the Coherence theorem for monoidal categories and the Strictification theorem for monoidal categories.

Further terminology

Let be a monoid category.

• Iff all the natural isomorphisms are the identity natural transformation, then is said to be a Strict monoidal category, which is a Monoid object in Category of small categories.
• Iff is the categorical product then is said to be a Cartesian category.
• Iff has a right adjoint internal hom-functor in a compatible way it is a Closed monoidal category.

The appropriate morphism of monoidal categories is the Monoidal functor, which allows the definition of the Category of monoidal categories.

Properties

• One can define a Monoid object

Other perspectives

A monoidal category may be viewed as

• A single-object bicategory

Diagrammatics

The diagrammatics of a monoidal category are single faced string diagrams in dimensions.

See also

• Examples of monoidal categories
• Monoidal functor
• Monoidal category

---

develop | en | sembr

Footnotes

1. 1978. Categories for the working mathematician ↩