Category of quivers

The category of quivers $𝖰 𝗎 𝗂 𝗏$ is a category where an object is a quiver and a morphism is a quiver homomorphism. #m/def/cat It is therefore identical to the category of presheaves $𝖯 𝖲 𝗁 (Θ 2 ―――)$ on the 2-Kronecker category.

Representable quivers

Viewing $𝖰 𝗎 𝗂 𝗏$ as a category of presheaves, we get two representable quivers $よ 0$ and $よ 1$ , “the walking vertex” and “the walking edge” respectively, since by the Yoneda lemma, for any quiver $Γ$ ,

Γ 0 ≅ 𝖰 𝗎 𝗂 𝗏 (よ 0, Γ), Γ 1 ≅ 𝖰 𝗎 𝗂 𝗏 (よ 1, Γ) .

Of course this is easily seen without the Yoneda lemma: If we draw the corresponding quivers, we get

so $よ 0 ≅ ⃗ 𝐴 1$ and $よ 1 ≅ Θ 1$ .

Important functors

There is a natural way $𝖰 𝗎 𝗂 𝗏 \to 𝖦 𝗋 𝖺 𝗉 𝗁$ to associate every quiver to an “underlying” general graph, see Equivalence of quivers and general graphs.
This factorizes the forgetful functor $V$ into the underlying vertex set.
See Underlying quiver and Free category

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