Quiver

A quiver is, loosely speaking, a (strict) category minus the algebra — the oidification of a set. A quiver¹ $Γ$ consists of #m/def/quiv

A set $Γ 0$ of vertices;
For any pair of vertices, a set $Γ (𝑣, 𝑤)$ of edges.

These data are conveniently packaged as a presheaf $Γ : Θ 2 ――― 𝐨 𝐩 \to 𝖲 𝖾 𝗍$ on the 2-Kronecker category $Θ 2 ―――$ , also called the walking quiver: We take $Γ 0 = Γ (0)$ , and if $𝑠$ and $𝑡$ , for source and target, are the non-identity morphisms of $Θ 2 ―――$ then

Γ (𝑣, 𝑤) = {𝑒 \in Γ 1 : Γ 𝑠 (𝑒) = 𝑣, Γ 𝑡 (𝑒) = 𝑤}

Further terminology

A morphism of the free category $Γ ――$ is called a path of $Γ$ .

Particular quivers

$𝑛$ -Kronecker quiver $Θ 𝑛$ .

#state/tidy | #lang/en | #SemBr

A calque of German Köcher. ↩

Quiver

Further terminology

See also

Particular quivers

Footnotes