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  "body": "\n## Definition\n\nA *quiver* is a quadruple $Q = (Q_0, Q_1, s, t)$ where:\n\n1. $Q_0$ is a set, whose elements are the *vertices*;\n2. $Q_1$ is a set, whose elements are the *arrows*;\n3. $s, t \\colon Q_1 \\to Q_0$ are maps, sending an arrow $a$ to its *source* $s(a)$ and its *target* $t(a)$.\n\nOne writes $a \\colon i \\to j$ when $s(a) = i$ and $t(a) = j$.\nThe quiver is *finite* if $Q_0$ and $Q_1$ are finite.\n\n## Remarks\n\nExample: the Kronecker quiver has $Q_0 = \\set{1, 2}$, $Q_1 = \\set{a, b}$ and $s(a) = s(b) = 1$, $t(a) = t(b) = 2$.\nIt satisfies the definition, though it has two arrows with the same source and target; a simple [[directed graph|0KFYGSHHF5]] would not allow this.\nA loop is an arrow $a$ with $s(a) = t(a)$, and it is also allowed.\nNon-example: an undirected graph is not a quiver as it stands, since its edges have no source and target, so clause 3 fails.\nChoosing an orientation of each edge turns it into a quiver.\nA 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 $i$.\nIn Mathlib, `Quiver` is a class on a type of vertices, with a type of arrows for each pair of vertices.\n",
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