# characterization of isomorphisms of quivers

Let $Q=({Q}_{0},{Q}_{1},s,t)$ and ${Q}^{\prime}=({Q}_{0}^{\prime},{Q}_{1}^{\prime},{s}^{\prime},{t}^{\prime})$ be quivers. Recall, that a morphism $F:Q\to {Q}^{\prime}$ is an isomorphism^{} if and only if there is a morphism $G:{Q}^{\prime}\to Q$ such that $FG=\mathrm{Id}({Q}^{\prime})$ and $GF=\mathrm{Id}(Q)$, where

$$\mathrm{Id}(Q):Q\to Q$$ |

is given by $\mathrm{Id}(Q)=(\mathrm{Id}{(Q)}_{0},\mathrm{Id}{(Q)}_{1})$, where both $\mathrm{Id}{(Q)}_{0}$ and $\mathrm{Id}{(Q)}_{1}$ are the identities^{} on ${Q}_{0}$, ${Q}_{1}$ respectively.

Proposition^{}. A morphism of quivers $F:Q\to {Q}^{\prime}$ is an isomorphism if and only if both ${F}_{0}$ and ${F}_{1}$ are bijctions.

Proof. ,,$\Rightarrow $” It follows from the definition of isomorphism that ${F}_{0}{G}_{0}=\mathrm{Id}{({Q}^{\prime})}_{0}$ and ${G}_{0}{F}_{0}=\mathrm{Id}{(Q)}_{0}$ for some ${G}_{0}:{Q}_{0}^{\prime}\to {Q}_{0}$. Thus ${F}_{0}$ is a bijection. The same argument is valid for ${F}_{1}$.

,,$\Leftarrow $” Assume that both ${F}_{0}$ and ${F}_{1}$ are bijections and define $G:{Q}_{0}^{\prime}\to {Q}_{0}$ and $H:{Q}_{1}^{\prime}\to {Q}_{1}$ by

$$G={F}_{0}^{-1},H={F}_{1}^{-1}.$$ |

Obviously $(G,H)$ is ,,the inverse^{}” of $F$ in the sense, that the equalites for compositions^{} hold. What is remain to prove is that $(G,H)$ is a morphism of quivers. Let $\alpha \in {Q}_{1}^{\prime}$. Then there exists an arrow $\beta \in {Q}_{1}$ such that

$${F}_{1}(\beta )=\alpha .$$ |

Thus

$$H(\alpha )=\beta .$$ |

Since $F$ is a morphism of quivers, then

$${s}^{\prime}(\alpha )={s}^{\prime}({F}_{1}(\beta ))={F}_{0}(s(\beta )),$$ |

which implies that

$$G({s}^{\prime}(\alpha ))=s(\beta )=s(H(\alpha )).$$ |

The same arguments hold for the target function $t$, which completes^{} the proof. $\mathrm{\square}$

Title | characterization of isomorphisms of quivers |
---|---|

Canonical name | CharacterizationOfIsomorphismsOfQuivers |

Date of creation | 2013-03-22 19:17:31 |

Last modified on | 2013-03-22 19:17:31 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Theorem |

Classification | msc 14L24 |