Monoidal internalization

Monoid object

Let $(𝖢, \otimes, 𝟙, 𝛼, 𝜆, 𝜌)$ be a monoidal category. A monoid in $𝖢$ consists of the data #m/def/cat

𝟙 𝜂 \to 𝑀 𝜇 \leftarrow 𝑀 \otimes 𝑀

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

and the associative law.

We can thence define a Monoid morphism and $𝖬 𝗈 𝗇 𝖢$ .

Commutative monoid

If $𝖢$ is symmetric, a comonoid satisfying the commutative law

is called commutative.

Properties

As in the traditional case, there exists at most one unit $𝑒 : 𝟙 \to 𝑀$ compatible with the multiplication $𝑚 : 𝑀 \otimes 𝑀 \to 𝑀$ .

Examples

A monoid in $𝖲 𝖾 𝗍$ with the cartesian product is a regulat monoid.
A monoid in $𝖠 𝖻$ is a ring.
More generally, for a commutative ring $𝑅$ , a monoid in $𝑅 𝖬 𝗈 𝖽$ is an $K$ -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.

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