Measuring coalgebra
= Measuring coalgebra
{title2=$p(c)(ab)=\sum p(c_{(1)})(a)p(c_{(2)})(b)$}
For unital <algebras over a field> $A,B$, a <coalgebra> $C$ measures $A$ into $B$ when a linear map $p:C\to\operatorname{Hom}_k(A,B)$ satisfies the displayed identity and $p(c)(1_A)=\varepsilon(c)1_B$. <Group-like elements> therefore act by unital <algebra homomorphisms over a field>.