Coïdeal

Let $C$ be a $K$ -comonoid. A $K$ -submodule $𝐼 \leq K C$ is said to be a coïdeal iff #m/def/hopf

Δ (𝐼) \leq K 𝐼 \otimes C + C \otimes 𝐼 .

and $𝜖 (𝐼) = 0$ .

Motivation

It may be unclear in what sense the above definition is dual to that of a (two-sided) ideal of a $K$ -monoid. The analogy is made clearer when the defining property of the latter is written

𝜇 (𝐼 \otimes C + C \otimes 𝐼) \leq 𝐼 .

The kernel of a $K$ -comonoid morphism is a coïdeal.

#state/tidy | #lang/en | #SemBr