coalgebra

Definition

A coalgebra is a vector space over a field equipped with a comultiplication map

and a counit map

These maps must satisfy the conditions

1. Coassociativity law:

2. Counit law:

These conditions are easier to see in diagram form:

Coassociativity:

Counit:

Dual to associative algebra

A coalgebra is a sort of dual construction to that of an associative algebra. Note that the maps above are the same as those for an associative algebra but “with the arrows going the other way”.