Coalgebra

This article is about a standard (though not very rudimentary) definition in an area related to, but not strictly part of, group theory

This article or section of article is sourced from:Wikipedia

Definition

A coalgebra is the following data:

$K$ a field and $C$ be a $K$-vector space, equipped with the following data:

• A linear map $\Delta :C\to C\otimes _{K}C$
• A linear map $\epsilon :C\to K$

satisfying the following compatibility conditions:

• Coassociativity:

$(id_{C}\otimes \Delta )\circ \Delta =(\Delta \otimes id_{C})\circ \Delta$

• Counit:

$(id_{C}\otimes \epsilon )\circ \Delta =id_{C}=(\epsilon \otimes id_{C})\circ \Delta$