\begin{definition}[Definition:Path (Category Theory)]
Let $\GG$ be a graph.
A (non-empty) '''path''' in $\GG$ is a (finite) sequence $\tuple {i, a_1, \ldots, a_n, j}$ such that:
:$n > 1$
:$a_1, \ldots, a_n$ are edges of $\GG$
:$i$ is the source of $a_1$
:$j$ is the destination of $a_n$
:For $1 \le k < n$ the destination of $a_k$ is the source of $a_{k + 1}$
This is usually written in the form:
:$i \stackrel {a_1} \longrightarrow * \stackrel {a_2} \longrightarrow \cdots \stackrel {a_{n - 1} } \longrightarrow * \stackrel {a_n} \longrightarrow j$
For any vertex $i$ the '''empty path''' from $i$ to $i$ is the pair $\tuple {i, i}$.
Paths are '''composed''' by concatenation:
:$\tuple {j, b_1, \ldots, b_n, k} \tuple {i, a_1, \ldots, a_m, j} = \tuple {i, a_1, \ldots, a_m, b_1, \ldots, b_n, k}$
Here we have used the somewhat awkward but more common right-to-left notation for composition.
Category:Definitions/Category Theory
\end{definition}