Category theory MOC

Monoidal category

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

1. a functor ( ⊗) :𝖢 ×𝖢 →𝖢 called the tensor product;
2. an object 1 ∈𝖢 called the tensor unit;
3. a natural isomorphism with components 𝛼𝑥,𝑦,𝑧 :(𝑥 ⊗𝑦) ⊗𝑧 →𝑥 ⊗(𝑦 ⊗𝑧) in [[Functor category|𝖢𝖢×𝖢×𝖢]] called the associator;
4. a natural isomorphism with components 𝜆𝑥 :1 ⊗𝑥 →𝑥 in [[Endofunctor category|𝖢𝖢]] called the left-unitor; and
5. a natural isomorphism with components 𝜌𝑥 :𝑥 ⊗1 →𝑥 in [[Endofunctor category|𝖢𝖢]] 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 (𝖢, ⊗,𝛼,𝜆,𝜌) be a monoid category.

• Iff all the natural isomorphisms 𝛼,𝜆,𝜌 are the identity natural transformation, then 𝖢 is said to be a Strict monoidal category, which is a Monoid object in Category of small categories.
• Iff ( ⊗) is the categorical product then 𝖢 is said to be a Cartesian category.
• Iff 𝖢 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 1 +1 dimensions.

See also

• Examples of monoidal categories
• Monoidal functor
• Monoidal category

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Footnotes

1. 1978. Categories for the working mathematician ↩