# fundamental isomorphism theorem for coalgebras

Let $(C,\mathrm{\Delta},\epsilon )$ and $(D,{\mathrm{\Delta}}^{\prime},{\epsilon}^{\prime})$ be coalgebras. Recall, that if ${D}_{0}\subseteq D$ is a subcoalgebra, then $({D}_{0},{\mathrm{\Delta}}_{|{D}_{0}}^{\prime},{\epsilon}_{|{D}_{0}}^{\prime})$ is a coalgebra. On the other hand, if $I\subseteq C$ is a coideal, then there is a canonical coalgebra structure^{} on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details).

Theorem. If $f:C\to D$ is a coalgebra homomorphism, then $\mathrm{ker}(f)$ is a coideal, $\mathrm{im}(f)$ is a subcoalgebra and a mapping ${f}^{\prime}:C/\mathrm{ker}(f)\to \mathrm{im}(f)$ defined by ${f}^{\prime}\left(c+\mathrm{ker}(f)\right)=f(c)$ is a well defined coalgebra isomorphism^{}.

Title | fundamental isomorphism theorem for coalgebras |
---|---|

Canonical name | FundamentalIsomorphismTheoremForCoalgebras |

Date of creation | 2013-03-22 18:49:30 |

Last modified on | 2013-03-22 18:49:30 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Theorem |

Classification | msc 16W30 |