Quiver

Remarks

Example: the Kronecker quiver has $Q_0 = \set{1, 2}$, $Q_1 = \set{a, b}$ and $s(a) = s(b) = 1$, $t(a) = t(b) = 2$. It satisfies the definition, though it has two arrows with the same source and target; a simple directed graph would not allow this. A loop is an arrow $a$ with $s(a) = t(a)$, and it is also allowed. Non-example: an undirected graph is not a quiver as it stands, since its edges have no source and target, so clause 3 fails. Choosing an orientation of each edge turns it into a quiver. A path of length $n \ge 1$ is a sequence of arrows $a_1, \dots, a_n$ with $t(a_r) = s(a_{r+1})$; there is also a trivial path $e_i$ at each vertex $i$. In Mathlib, Quiver is a class on a type of vertices, with a type of arrows for each pair of vertices.