Quiver homomorphism

A homomorphism $𝜂 : Γ 1 \to Γ 2$ of quivers \Gamma_{1}, \Gamma_{2} : \cat Q \to \Set is just a natural transformation of the corresponding functors, cat i.e. a pair of functions $𝜂 𝑉 : Γ 1 𝑉 \to Γ 2 𝑉$ and $𝜂 𝐸 : Γ 1 𝐸 \to Γ 2 𝐸$ mapping vertices and edges respectively such that

(Γ 2 𝑠) (𝜂 𝐸 (𝑎)) = 𝜂 𝑉 ((Γ 1 𝑠) (𝑎)) (Γ 2 𝑡) (𝜂 𝐸 (𝑎)) = 𝜂 𝑉 ((Γ 1 𝑡) (𝑎))

for all $𝑎 \in Γ 1 𝐸$ , or equivalently

𝜂 𝐸 (Γ 1 (𝑣, 𝑤)) \subseteq Γ 2 (𝜂 𝑉 (𝑣), 𝜂 𝑉 (𝑤))

for all $𝑣, 𝑤 \in Γ 1 𝑉$ .

Proof these are equivalent

Suppose ^H1 holds. Let $𝑏 \in 𝜂 𝐸 (Γ 1 (𝑣, 𝑤))$ , i.e. $𝑏 = 𝜂 𝐸 (𝑎)$ for some $𝑎 \in Γ 1 𝐸$ such that $(Γ 1 𝑠) (𝑎) = 𝑣$ and $(Γ 2 𝑠) (𝑎) = 𝑤$ , in which case
$(Γ 2 𝑠) (𝑏) = 𝜂 𝑉 ((Γ 1 𝑠) (𝑎)) = 𝜂 𝑉 (𝑣) (Γ 2 𝑡) (𝑏) = 𝜂 𝑉 ((Γ 1 𝑡) (𝑎)) = 𝜂 𝑉 (𝑤)$
and thus $𝑏 \in Γ 2 (𝜂 𝑉 (𝑣), 𝜂 𝑉 (𝑤))$ . Therefore $𝜂 𝐸 (Γ 1 (𝑣, 𝑤)) \subseteq Γ 2 (𝜂 𝑉 (𝑣), 𝜂 𝑉 (𝑤))$ .

Now suppose ^H2 holds. Let $𝑎 \in Γ 1 𝐸$ , $𝑣 = (Γ 1 𝑠) (𝑎)$ , and $𝑤 = (Γ 2 𝑡) (𝑎)$ . Then $𝑎 \in Γ 1 (𝑣, 𝑤)$ , so $𝜂 𝐸 (𝑎) \in Γ 2 (𝜂 𝑉 (𝑣), 𝜂 𝑉 (𝑤))$ , whence
$(Γ 2 𝑠) (𝜂 𝐸 (𝑎)) = 𝜂 𝑉 (𝑣) = 𝜂 𝑉 ((Γ 1 𝑠) (𝑎)) (Γ 2 𝑡) (𝜂 𝐸 (𝑎)) = 𝜂 𝑉 (𝑤) = 𝜂 𝑉 ((Γ 1 𝑡) (𝑎))$
as required.

These form the morphisms in Category of quivers.

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Quiver homomorphism

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