Kerodon

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

Warning 5.5.3.8. There is an evident simplicial functor from the category $\operatorname{Kan}_{\ast }$ of pointed Kan complexes to the category $\operatorname{Kan}$ of Kan complexes, given on objects by the construction $(X,x) \mapsto X$. Passing to homotopy coherent nerves, we obtain a functor of $\infty $-categories $U: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}}$. Beware that the functor $U$ is not a left fibration of simplicial sets. For example, suppose we are given a $2$-simplex $\sigma $ of $\operatorname{\mathcal{S}}$, corresponding to a diagram 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 commutes up to a homotopy $\mu : (g \circ f) \rightarrow h$ (see Remark 5.5.1.3). Pick a vertex $x \in X$ and set $y = f(x)$ and $z = h(x)$, so that we have morphisms of pointed Kan complexes Kan complexes $f: (X,x) \rightarrow (Y,y)$ and $h: (X,x) \rightarrow (Z,z)$. This data determines a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{0} \ar [r]^-{ (\bullet , h, f) } \ar [d] & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \ar [d]^{U} \\ \Delta ^{2} \ar [r]^-{\sigma } \ar@ {-->}[ur] & \operatorname{\mathcal{S}}, } \]

which admits a solution if and only if $\mu (x): g(y) \rightarrow z$ is a degenerate edge of the Kan complex $Z$ (in which case $g(y) = z$, so that $g: (Y,y) \rightarrow (Z,z)$ is also a morphism of pointed Kan complexes).