Definition:Path (Category Theory)

Definition
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.