morphisms between quivers

Recall that a quadruple Q=(Q0,Q1,s,t) is a quiver, if Q0 is a set (whose elements are called vertices), Q1 is also a set (whose elements are called arrows) and s,t:Q1Q0 are functions which take each arrow to its source and target respectively.

Definition. A morphism from a quiver Q=(Q0,Q1,s,t) to a quiver Q=(Q0,Q1,s,t) is a pair


such that F0:Q0Q0, F1:Q1Q1 are functions which satisfy


In this case we write F:QQ. In other words F:QQ is a morphism of quivers, if for an arrow


in Q the following


is an arrow in Q.

If F:QQ and G:QQ′′ are morphisms between quivers, then we have the compositionMathworldPlanetmath


defined by


It can be easily checked, that GF 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 isomorphismPlanetmathPlanetmathPlanetmath. It can be shown, that two quivers Q, Q are isomorphic if and only if there exists a morphism of quivers


such that both F0 and F1 are bijections.

For example quivers


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