# coalgebra homomorphism

Let $(C,\Delta,\varepsilon)$ and $(D,\Delta^{\prime},\varepsilon^{\prime})$ be coalgebras.

Definition. Linear map $f:C\to D$ is called coalgebra homomorphism if $\Delta^{\prime}\circ f=(f\otimes f)\circ\Delta$ and $\varepsilon^{\prime}\circ f=\varepsilon$.

Examples. $1)$ Of course, if $D$ is a subcoalgebra of $C$, then the inclusion $i:D\to C$ is a coalgebra homomorphism. In particular, the identity is a coalgebra homomorphism.

$2)$ If $(C,\Delta,\varepsilon)$ is a coalgebra and $I\subseteq C$ is a coideal, then we have canonical coalgebra structur on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details). Then the projection $\pi:C\to C/I$ is a coalgebra homomorphism. Furthermore, one can show that the canonical coalgebra structure on $C/I$ is a unique coalgebra structure such that $\pi$ is a coalgebra homomorphism.

Title coalgebra homomorphism CoalgebraHomomorphism 2013-03-22 18:49:25 2013-03-22 18:49:25 joking (16130) joking (16130) 4 joking (16130) Definition msc 16W30