Comonoid object

Let $(𝖢, \otimes, 1, 𝛼, 𝜆, 𝜌)$ be a monoidal category. A comonoid in $𝖢$ is a monoid in the opposite category $𝖢 𝐨 𝐩$ , 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

the coässociative law,

and optionally the cocommutative law (whence it is called cocommutative).

The category of comonoid objects is Category of comonoid objects, which is simply [[Category of monoid objects| $𝖬 𝗈 𝗇 𝖢 𝐨 𝐩$ ]].

Higher comultiplications

Note that by coässociativity, we can unambiguously define

Δ 𝑛 : 𝑀 \to 𝑀 \otimes (𝑛 + 1)

Examples

A comonoid in Category of monoid objects 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 R-comonoid.

Table of Contents

Comonoid object

Higher comultiplications

Examples

See also

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