# morphisms between quivers

Recall that a quadruple $Q=({Q}_{0},{Q}_{1},s,t)$ is a quiver, if ${Q}_{0}$ is a set (whose elements are called vertices), ${Q}_{1}$ is also a set (whose elements are called arrows) and $s,t:{Q}_{1}\to {Q}_{0}$ are functions which take each arrow to its source and target respectively.

Definition. A morphism from a quiver $Q=({Q}_{0},{Q}_{1},s,t)$ to a quiver ${Q}^{\prime}=({Q}_{0}^{\prime},{Q}_{1}^{\prime},{s}^{\prime},{t}^{\prime})$ is a pair

$$F=({F}_{0},{F}_{1})$$ |

such that ${F}_{0}:{Q}_{0}\to {Q}_{0}^{\prime}$, ${F}_{1}:{Q}_{1}\to {Q}_{1}^{\prime}$ are functions which satisfy

$${s}^{\prime}\left({F}_{1}(\alpha )\right)={F}_{0}\left(s(\alpha )\right);$$ |

$${t}^{\prime}\left({F}_{1}(\alpha )\right)={F}_{0}\left(t(\alpha )\right).$$ |

In this case we write $F:Q\to {Q}^{\prime}$. In other words $F:Q\to {Q}^{\prime}$ is a morphism of quivers, if for an arrow

$$\text{xymatrix}x\text{ar}{[r]}^{\alpha}\mathrm{\&}y$$ |

in $Q$ the following

$$\text{xymatrix}{F}_{0}(x)\text{ar}{[r]}^{{F}_{1}(\alpha )}\mathrm{\&}{F}_{0}(y)$$ |

is an arrow in ${Q}^{\prime}$.

If $F:Q\to {Q}^{\prime}$ and $G:{Q}^{\prime}\to {Q}^{\prime \prime}$ are morphisms between quivers, then we have the composition^{}

$$G\circ F:Q\to {Q}^{\prime \prime}$$ |

defined by

$$G\circ F=({G}_{0}\circ {F}_{0},{G}_{1}\circ {F}_{1}).$$ |

It can be easily checked, that $G\circ F$ is again a morphism between quivers.

The class of all quivers, all morphisms between together with the composition is a category. In particular we have a notion of isomorphism^{}. It can be shown, that two quivers $Q$, ${Q}^{\prime}$ are isomorphic if and only if there exists a morphism of quivers

$$F:Q\to {Q}^{\prime}$$ |

such that both ${F}_{0}$ and ${F}_{1}$ are bijections.

For example quivers

$$\text{xymatrix}Q:1\text{ar}[r]\mathrm{\&}2\mathrm{\&}\mathrm{\&}\mathrm{\&}{Q}^{\prime}:1\mathrm{\&}2\text{ar}[l]$$ |

are isomorphic, although not equal.

Title | morphisms between quivers |
---|---|

Canonical name | MorphismsBetweenQuivers |

Date of creation | 2013-03-22 19:16:57 |

Last modified on | 2013-03-22 19:16:57 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 5 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 14L24 |