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Warning 1.2.5.7. Let $S_{\bullet }$ be a simplicial set. Our proof of Proposition 1.2.5.4 gives a construction of the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ by generators and relations. The result of this construction is not easy to describe. If $x$ and $y$ are vertices of $S_{\bullet }$, then every morphism from $x$ to $y$ in $\mathrm{h} \mathit{S}_{\bullet }$ can be represented by a composition

\[ [e_ n] \circ [e_{n-1}] \circ \cdots \circ [e_1], \]

where $\{ e_ i \} _{0 \leq i \leq n}$ is a sequence of edges satisfying

\[ d_1(e_1) = x \quad \quad d_0(e_ i) = d_1(e_{i+1}) \quad \quad d_0( e_ n ) = y. \]

In general, it can be difficult to determine whether or not two such compositions represent the same morphism of $\mathrm{h} \mathit{S}_{\bullet }$ (even for finite simplicial sets, this question is algorithmically undecidable). However, there are two situations in which the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ admits a simpler description:

  • Let $S_{\bullet }$ be a simplicial set of dimension $\leq 1$, which we can identify with a directed graph $G$ (Proposition 1.1.5.9). In this case, the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ 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 1.2.6.5) which we study in §1.2.6.

  • Let $S_{\bullet }$ be an $\infty $-category. In this case, every morphism in the homotopy category $\operatorname{\mathcal{C}}= \mathrm{h} \mathit{S}_{\bullet }$ can be represented by a single edge of $S_{\bullet }$, rather than a composition of edges (in other words, the canonical map $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ is surjective on edges), and two edges of $S_{\bullet }$ represent the same morphism in $\mathrm{h} \mathit{S}_{\bullet }$ if and only if they are homotopic (Definition 1.3.3.1). This leads to a more explicit description of the homotopy category $\operatorname{\mathcal{C}}$ (generalizing Exercise 1.2.5.2) which we will discuss in §1.3.5 (see Proposition 1.3.5.7).