Category theory MOC

Monoidal category

A monoidal category is the vertical categorification of a monoid. Explicitly, a monoidal category 𝖢 is equipped with¹ #m/def/cat

1. a bifunctor ( ⊗) :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 𝖢𝖢×𝖢×𝖢 called the associator;
4. a natural isomorphism with components 𝜆𝑥 :1 ⊗𝑥 →𝑥 in 𝖢𝖢 called the left-unitor; and
5. a natural isomorphism with components 𝜌𝑥 :𝑥 ⊗1 →𝑥 in 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 canonical natural isomorphism, by the Coherence theorem for monoidal categories.

Further terminology

Let (𝖢, ⊗,𝛼,𝜆,𝜌) be a monoid category.

• 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.

Other perspectives

A monoidal category may be viewed as a single-object (“connected”) 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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