Kerodon

Example 1.3.7.3. Let $G$ be a directed graph having a single vertex $\operatorname{Vert}(G) = \{ x \} $. Then the path category $\operatorname{Path}[G]$ has a single object $x$, and can therefore be identified with the category $BM$ associated to the monoid $M = \operatorname{End}_{\operatorname{Path}[G]}(x) = \operatorname{Hom}_{\operatorname{Path}[G]}(x,x)$ (see Construction 1.3.2.5). Note that the elements of $M$ can be identified with (possibly empty) sequences of elements of the set $\operatorname{Edge}(G)$, and that the multiplication on $M$ is given by concatenation of sequences. In other words, $M$ can be identified with the free monoid generated by the set $\operatorname{Edge}(M)$ (this identification is not completely tautological: it can be regarded as a special case of Proposition 1.3.7.5 below).