Comonoid object

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

1 𝜖 \leftarrow 𝑀 Δ ⟶ 𝑀 \otimes 𝑀

where $𝜖$ is called the coünit and $Δ$ is called the comultiplication, and these satisfy the left/right coünit laws

and the coässociative law.

The category of comonoid objects is $𝖢 𝗈 𝗆 𝗈 𝗇 𝖢$ , which is simply the opposite category of $𝖬 𝗈 𝗇 𝖢 𝐨 𝐩$ .

Cocommutative cocomonoid

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

is called cocommutative.

Higher comultiplications

Note that by coässociativity, we can unambiguously define

Δ 𝑛 : 𝑀 \to 𝑀 \otimes (𝑛 + 1)

Examples

A comonoid in $𝖬 𝗈 𝗇 𝖢$ is a Bimonoid object (which is equivalently a monoid in $𝖬 𝗈 𝗇 𝖢 𝐨 𝐩$ ).
A comonoid in $𝑅 𝖬 𝗈 𝖽$ is an $K$ -comonoid.

Comonoid object

Cocommutative cocomonoid

Higher comultiplications

Examples

See also