Kerodon

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

Remark 5.5.1.3. Let $\operatorname{N}_{\bullet }(\operatorname{Kan})$ denote the nerve of the category of Kan complexes, where we view $\operatorname{Kan}$ as an ordinary category. There is an evident monomorphism of simplicial sets

\[ \iota : \operatorname{N}_{\bullet }(\operatorname{Kan}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}, \]

which is bijective on simplices of dimension $\leq 1$ (Example 2.4.3.9). In other words:

  • The objects of the $\infty $-category $\operatorname{\mathcal{S}}$ are Kan complexes.

  • If $X$ and $Y$ are Kan complexes, then morphisms $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{S}}$ can be identified with morphisms of simplicial sets from $X$ to $Y$.

However, $\iota $ is not bijective on simplices of dimension $\geq 2$. For example, $2$-simplices of $\operatorname{\mathcal{S}}$ can be identified with diagrams of Kan complexes

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

which commute up to a specified homotopy $\mu : (g \circ f) \rightarrow h$.