# coalgebra homomorphism

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

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

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 $(C,\mathrm{\Delta},\epsilon )$ 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 |
---|---|

Canonical name | CoalgebraHomomorphism |

Date of creation | 2013-03-22 18:49:25 |

Last modified on | 2013-03-22 18:49:25 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 16W30 |