Internalization

Semigroup object

Let be a monoidal category. A semigroup in consists of the data cat

where is called the multiplication, and these satisfy the associative law. Moreover, if we are in a Symmetric monoidal category with braiding , then 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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