Definition 1.5.2.5. Let $K$ be a simplicial set of dimension $\leq 1$, which we will identify with a directed graph $G$ (see Proposition 1.1.6.9). Assume that $G$ satisfies the following additional conditions:
- $(a)$
For every pair of vertices $v,w \in \operatorname{Vert}(G)$, there is at most one edge of $G$ with source $v$ and target $w$. We will denote this edge (if it exists) by $(v,w) \in \operatorname{Edge}(G)$.
- $(b)$
The graph $G$ has no directed cycles. That is, if there exists a sequence of vertices $v_0, v_1, \ldots , v_ n \in \operatorname{Vert}(G)$ with the property that the edges $( v_{i-1}, v_ i)$ exist for $1 \leq i \leq n$, then either $n = 0$ or $v_0 \neq v_ n$.
Let $\operatorname{\mathcal{C}}$ be an ordinary category and suppose we are given a diagram $\sigma : K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we identify with a pair $( \{ C_ v \} _{v \in \operatorname{Vert}(G) }, \{ f_{w,v}: C_ v \rightarrow C_ w \} _{(v,w) \in \operatorname{Edge}(G)} )$. We will say that the diagram $\sigma $ commutes (or that $\sigma $ is a commutative diagram) if the following additional condition is satisfied:
- $(c)$
Let $v$ and $w$ be vertices of $G$ which are joined by directed paths $( v = v_0, v_1, \ldots , v_ m = w)$ and $(v = v'_0, v'_1, \ldots , v'_ n = w)$ (so that the edges $( v_{i-1} , v_ i ), ( v'_{j-1}, v'_ j) \in \operatorname{Edge}(G)$ exist for $1 \leq i \leq m$ and $1 \leq j \leq n$). Then we have an identity
\[ f_{ v_{m}, v_{m-1} } \circ f_{ v_{m-1}, v_{m-2} } \circ \cdots \circ f_{ v_1, v_0} = f_{ v'_{n}, v'_{n-1} } \circ f_{ v'_{n-1}, v'_{n-2} } \circ \cdots \circ f_{ v'_1, v'_0} \]in the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_{v}, C_{w} )$.