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

satisfying the following compatibility conditions:

• Coassociativity:

$(i{d}_C\otimes \mathrm{\Delta })\circ \mathrm{\Delta }=(\mathrm{\Delta }\otimes i{d}_C)\circ \mathrm{\Delta }$

• Counit:

$(i{d}_C\otimes ϵ)\circ \mathrm{\Delta }=i{d}_C=(ϵ\otimes i{d}_C)\circ \mathrm{\Delta }$