$K$ -comonoid

Let $K$ be a commutative ring. A $K$ -comonoid $A$ is a comonoid in $K 𝖬 𝗈 𝖽$ , #m/def/ralg/hopf and thus consists of the data

1 𝜖 \leftarrow A Δ ⟶ A \otimes A

satisfying the coünit law and the coässociative law.

Sweedler notation

It is convenient to introduce Sweedler notation, where we write

Δ 𝑎 = \sum (𝑎) 𝑎 (1) \otimes 𝑎 (2) .

This extends to higher comultiplications, so that

Δ 𝑛 𝑎 = \sum (𝑎) 𝑎 (1) \otimes \dots \otimes 𝑎 (𝑛) .

The idea is that the tensor $Δ 𝑎$ may be decomposed into a finite sum of separable tensors, so we feel free to invoke such a decomposition without fixing it explicitly.

Results

Dual $K$ -monoid of a $K$ -comonoid

Examples

Free $K$ -comonoid

K-comonoid

Sweedler notation

Results

Examples

See also

$K$ -comonoid