Example 1

Let C be a quiver. Let Ch(C) denote the set of paths in C and $o:$ Ch(C) $\rightarrow$ Som(C)$,t:$ Ch(C) $\rightarrow$ Som(C) the mappings which associate with a path its origin and its end. The quadruple $\Omega_C=$ (Som(C), Ch(C)$, o, t$) is a quiver. Endow it with the law of composition defined by juxtaposition of paths. This law of composition is associative; for every vertex $a$ of C, the constant loop $e_a= (a)$ of origin $a$ is an identity element at $a$. Thus, $\Omega_C$ is a category. It is said to be the category of paths of the quiver C.