Internalization

Monoid object

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

𝟙𝑒→𝑀𝑚⟵𝑀⊗𝑀

where 𝑒 is called the unit and 𝑚 is called the multiplication, and these satisfy the left/right unit laws, and 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

Unit laws:

Associative law:

Commutative law:

String diagrams

Unitality:

Associativity:

Commutativity:

We can thence define a Homomorphism of monoid objects and Category of monoid objects. These concepts admit duals, see Comonoid object. See also the weakening of Semigroup object.

Properties

• As in the traditional case, there exists at most one unit 𝑒 :𝟙 →𝑀 compatible with the multiplication 𝑚 :𝑀 ⊗𝑀 →𝑀.

Examples

• A monoid in Category of sets with the cartesian product is a regulat monoid.
• A monoid in Category of abelian groups is a ring.
• More generally, for a commutative ring 𝑅, a monoid in Category of left modules is an R-monoid.
• A monoid in an Endofunctor category is a Monad.

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