Monoidal internalization

Monoid object

Let be a monoidal category. A monoid in consists of the data #m/def/cat

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

and the associative law.

We can thence define a Monoid morphism and .

Commutative monoid

If is symmetric, a comonoid satisfying the commutative law

is called commutative.

Properties

• As in the traditional case, there exists at most one unit compatible with the multiplication .

Examples

• A monoid in with the cartesian product is a regulat monoid.
• A monoid in is a ring.
• More generally, for a commutative ring , a monoid in is an -monoid.
• A monoid in an Endofunctor category is a Monad.

See also

• These concepts admit duals, see Comonoid object.
• See also the weakening of Semigroup object.

#state/develop | #lang/en | #SemBr