Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.4.6.10. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category, so that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1). Suppose we are given a pair of morphisms $f,g: X \rightarrow Y$ in the underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$ having the same source and target. Let $\sigma _0: \operatorname{\partial }\Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be the map corresponding to the (possibly noncommutative) diagram

\[ \xymatrix { & Y \ar [dr]^{ \operatorname{id}_ Y } & \\ X \ar [ur]^{g} \ar [rr]^{f} & & Y. } \]

Applying Proposition 2.4.6.9 we obtain a bijection from the set of homotopies from $g$ to $f$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (in the sense of Definition 1.3.3.1) to the set of homotopies from $f$ to $g$ in the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ (in the sense of Definition 2.4.1.6). In particular, we see that $f$ and $g$ are homotopic in $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if they are homotopic in $\operatorname{\mathcal{C}}_{\bullet }$.