Warning 1.3.6.7. Let $S$ be a simplicial set. The proof of Proposition 1.3.6.4 gives a construction of the homotopy category $\mathrm{h} \mathit{S}$ by generators and relations. The result of this construction is not always easy to describe. If $x$ and $y$ are vertices of $S$, then every morphism from $x$ to $y$ in $\mathrm{h} \mathit{S}$ can be represented by a composition
where $\{ e_ i \} _{0 \leq i \leq n}$ is a sequence of edges satisfying
In general, it can be difficult to determine whether or not two such compositions represent the same morphism of $\mathrm{h} \mathit{S}$ (even for finite simplicial sets, this question is algorithmically undecidable). However, there are two situations in which the homotopy category $\mathrm{h} \mathit{S}$ admits a simpler description:
Let $S$ be a simplicial set of dimension $\leq 1$, which we can identify with a directed graph $G$ (Proposition 1.1.6.9). In this case, the homotopy category $\mathrm{h} \mathit{S}$ is generated freely by the vertices and edges of the graph $G$: that is, it can be identified with the path category of $G$ (Proposition 1.3.7.5) which we study in §1.3.7.
Let $S$ be an $\infty $-category. In this case, every morphism in the homotopy category $\operatorname{\mathcal{C}}= \mathrm{h} \mathit{S}$ can be represented by a single edge of $S$, rather than a composition of edges (in other words, the canonical map $u: S \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ is surjective on edges), and two edges of $S$ represent the same morphism in $\mathrm{h} \mathit{S}$ if and only if they are homotopic (Definition 1.4.3.1). This leads to a more explicit description of the homotopy category $\operatorname{\mathcal{C}}$ (generalizing Exercise 1.3.6.2) which we will discuss in §1.4.5 (see Proposition 1.4.5.7).