Kerodon

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Remark 1.4.7.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category (Definition 1.3.5.3). Then the canonical map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an epimorphism of simplicial sets: that is, it induces a surjection on $n$-simplices for each $n \geq 0$. To prove this, we note that there is a commutative diagram

\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )) \ar [d]^{\sim } \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Spine}[n], \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Spine}[n], \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) ), } \]

where the left vertical map is surjective (Example 1.4.7.7) and the right vertical map is bijective (Remark 1.4.7.8). It therefore suffices to show that the bottom horizontal map is surjective: that is, every sequence of composable morphisms

\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \xrightarrow {f_3} X_3 \rightarrow \cdots \xrightarrow {f_ n} X_ n \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ can be lifted to a sequence of composable morphisms in $\operatorname{\mathcal{C}}$, which is immediate from the definition of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.