Kerodon

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

Example 5.4.3.4 (Morphisms of $\operatorname{\mathcal{S}}_{\ast }$). Let $(X,x)$ and $(Y,y)$ be pointed Kan complexes, regarded as objects of the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$. By definition, a morphism from $(X,x)$ to $(Y,y)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ can be identified with a $2$-simplex $\sigma $ of the simplicial set $\operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan})$, which we can identify with a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]_{f} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-60pt>^-{h} & \\ \Delta ^{0} \ar [ur]^{x} \ar [rr]_{y} & & Y } \]

which commutes up to a specified homotopy $h$. In other words, a morphism from $(X,x)$ to $(Y,y)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ can be identified with a pair $(f,h)$, where $f: X \rightarrow Y$ is a morphism of Kan complexes and $h: f(x) \rightarrow y$ is an edge of the simplicial set $Y$.