# coalgebra isomorphisms and isomorphic coalgebras

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

Definition. We will say that coalgebra homomorphism $f:C\to D$ is a coalgebra isomorphism^{}, if there exists a coalgebra homomorphism $g:D\to C$ such that $f\circ g={\mathrm{id}}_{D}$ and $g\circ f={\mathrm{id}}_{C}$.

Remark. Of course every coalgebra isomorphism is a linear isomorphism, thus it is ,,one-to-one” and ,,onto”. One can show that the converse^{} also holds, i.e. if $f:C\to D$ is a coalgebra homomorphism such that $f$ is ,,one-to-one” and ,,onto”, then $f$ is a coalgebra isomorphism.

Definition. We will say that coalgebras $(C,\mathrm{\Delta},\epsilon )$ and $(D,{\mathrm{\Delta}}^{\prime},{\epsilon}^{\prime})$ are isomorphic if there exists coalgebra isomorphism $f:C\to D$. In this case we often write $(C,\mathrm{\Delta},\epsilon )\simeq (D,{\mathrm{\Delta}}^{\prime},{\epsilon}^{\prime})$ or simply $C\simeq D$ if structure maps^{} are known from the context.

Remarks. Of course the relation^{} ,,$\simeq $” is an equivalence relation^{}. Furthermore, (from the coalgebraic point of view) isomorphic coalgebras are the same, i.e. they share all coalgebraic properties.

Title | coalgebra isomorphisms and isomorphic coalgebras |
---|---|

Canonical name | CoalgebraIsomorphismsAndIsomorphicCoalgebras |

Date of creation | 2013-03-22 18:49:28 |

Last modified on | 2013-03-22 18:49:28 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 16W30 |