Internalization

Semigroup object

Let (𝖢, ⊗,𝟙,𝛼,𝜆,𝜌) be a monoidal category. A semigroup in 𝖢 consists of the data cat

𝑀⊗𝑀𝑚⟶𝑀

where 𝑚 is called the multiplication, and these satisfy the associative law. Moreover, if we are in a Symmetric monoidal category with braiding 𝜏, then (𝑀,𝑚,𝑒) is called commutative iff it satisfies the commutative law.

Commutative diagrams

Associative law:

Commutative law:

String diagrams

Associativity:

Commutativity:

We can thence define a Homomorphism of semigroup objects and Category of semigroup objects. These concepts admit duals, see Cosemigroup object.

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