category of quivers is concrete


Let 𝒬 denote the categoryMathworldPlanetmath of all quivers and quiver morphismsMathworldPlanetmath with standard composition. If Q=(Q0,Q1,s,t) is a quiver, then we can associate with Q the set

S⁢(Q)=Q0⊔Q1

where ,,⊔” denotes the disjoint unionMathworldPlanetmathPlanetmath of sets.

Furthermore, if F:Q→Q′ is a morphism of quivers, then F induces function

S⁢(F):S⁢(Q)→S⁢(Q′)

by putting S⁢(F)⁢(a)=F0⁢(a) if a∈Q0 and S⁢(F)⁢(α)=F1⁢(α) if α∈Q1.

PropositionPlanetmathPlanetmathPlanetmath. The category 𝒬 together with S:𝒬→𝒮⁢ℰ⁢𝒯 is a concrete category over the category of all sets 𝒮⁢ℰ⁢𝒯.

Proof. The fact that S is a functorMathworldPlanetmath we leave as a simple exercise. Now assume, that F,G:Q→Q′ are morphisms of quivers such that S⁢(F)=S⁢(G). It follows, that for any vertex a∈Q0 and any arrow α∈Q1 we have

F0⁢(a)=S⁢(F)⁢(a)=S⁢(G)⁢(a)=G0⁢(a);
F1⁢(α)=S⁢(F)⁢(α)=S⁢(G)⁢(α)=G1⁢(α)

which clearly proves that F=G. This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. □

Remark. Note, that if F:Q→Q′ is a morphism of quivers, then F is injectivePlanetmathPlanetmath in (𝒬,S) (see this entry (http://planetmath.org/InjectiveAndSurjectiveMorphismsInConcreteCategories) for details) if and only if both F0, F1 are injective. The same holds if we replace word ,,injective” with ,,surjectivePlanetmathPlanetmath”.

Title category of quivers is concrete
Canonical name CategoryOfQuiversIsConcrete
Date of creation 2013-03-22 19:17:28
Last modified on 2013-03-22 19:17:28
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 14L24