Definition 1

A quiver is a quadruple $(S,F, o, t)$, where S and F are sets and $o,t$ are mappings of F into S.

Let $C = (S,F, o, t)$ be a quiver. The elements of S are called the vertices of C. The elements of F are called the arrows of C. Let $f$ be an arrow of C; the vertex $o(f)$ is called the origin or the source of $f$, the vertex $t(f)$ is called the end, or the target, of $f$; one also says that $f$ joins the vertex $o(f)$ to the vertex $t(f)$. The mappings $o$ and $t$ are called respectively the source and target mappings, or origin and end, of the quiver C.

© N. Bourbaki and Springer-Verlag Berlin Heidelberg 2 016

N. Bourbaki, Algebraic Topology, DOI 10.1007/978-3-662-49361-8_2 151

When C is a quiver, Som(C), Fl(C), $o_C$ and $t_C$ will denote respectively the set of vertices, the set of arrows, the source mapping and the target mapping of C. Let $a$ and $b$ be vertices of C; the set of arrows of C joining $a$ to $b$ is denoted by Fl$_{a,b}(C)$, or again $C_{a,b}$.

To make the arguments easier to understand, one may represent a quiver by a diagram composed of points and arrowed lines corresponding respectively to the vertices and the arrows. The diagram:

$\bullet \bullet \bullet k$ $abc$ $f_2h$

represents a quiver whose vertices are $a,b,c$, whose arrows are $f_1, f_2, g, h, k$, and whose source and target mappings are given by