Definition:Path (Category Theory)

Definition
Let $\mathcal G$ be a graph.

A (non-empty) path in $\mathcal G$ is a sequence $(i,a_1,\ldots,a_n,j)$ such that:
 * $n > 1$
 * $a_1,\ldots, a_n$ are edges of $\mathcal G$
 * $i$ is the source of $a_1$ and $j$ is the destination of $a_n$
 * For $1 \leq k < n$ the destination of $a_k$ is the source of $a_{k+1}$

This is usually written in the form:


 * $\displaystyle i \stackrel{a_1}{\longrightarrow} * \stackrel{a_2}{\longrightarrow} \cdots \stackrel{a_{n-1}}{\longrightarrow} * \stackrel{a_n}{\longrightarrow} j$

For any vertex $i$ the empty path from $i$ to $i$ is the pair $(i,i)$.

Paths are composed by concatenation:


 * $\displaystyle (j,b_1,\ldots,b_n,k)(i,a_1,\ldots,a_m,j) = (i,a_1,\ldots,a_m,b_1,\ldots,b_n,k)$

Here we have used the somewhat awkward but more common right-to-left notation for composition.