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Proposition 5.4.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ be a full simplicial subcategory of $\operatorname{\mathcal{C}}_{X/}$ with the following property:

$(\ast )$

For every pair of objects $(Y,f)$ and $(Z,g)$ of the simplicial category $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}_{X/}$, the morphism of simplicial sets $ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$ is a Kan fibration.

Then the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ is an $\infty $-category, and the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ of Construction 5.4.2.17 restricts to a fully faithful functor of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$.

Proof of Proposition 5.4.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing an object $X$, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}_{X/}$ be a full simplicial subcategory satisfying hypothesis $(\ast )$ of Proposition 5.4.2.22. For every pair of objects $(Y,f), (Z,g) \in \operatorname{\mathcal{E}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet }$ is the fiber of the Kan fibration

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \]

over the vertex $g$, and is therefore a Kan complex (Remark 3.1.1.9). Applying Theorem 2.4.5.1, we conclude that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ is an $\infty $-category. We wish to show that, for every pair of objects $(Y,f), (Z,g) \in \operatorname{\mathcal{E}}$ as above, the coslice comparison morphism $c'$ induces a homotopy equivalence of morphism spaces

\[ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) }( (Y,f), (Z,g) ) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/ } }( (Y,f), (Z,g) ). \]

By virtue of Proposition 4.6.6.9, this is equivalent to the requirement that $c'$ induces a homotopy equivalence $\rho : \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})}( (Y,f), (Z,g) ) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/} }( (Y,f), (Z,g) )$ of left-pinched morphism spaces.

Construction 4.6.7.3 supplies comparison maps

\[ \overline{\theta }: \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) }^{\mathrm{L}}( (Y,f), (Z,g) ) \]

\[ \theta _{Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(Y,Z) \quad \quad \theta _{X,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Z), \]

which are homotopy equivalences of Kan complexes by virtue of Theorem 4.6.7.5. Let us regard $f: X \rightarrow Y$ as an edge of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, and let $Q$ denote the fiber $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z\} $. Since the inclusion $\{ 1\} \hookrightarrow \Delta ^1$ is right anodyne, the restriction map $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ Y/}$ is a trivial Kan fibration (Proposition 4.3.6.10), and therefore restricts to a trivial Kan fibration

\[ \pi : Q \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ Y/} \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z \} = \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} (Y, Z). \]

In particular, $Q$ is a Kan complex and $\pi $ is a homotopy equivalence. Let $\pi '$ denote the restriction map

\[ Q \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ X/} \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z \} = \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} (X, Z). \]

Note that $\pi '$ is a pullback of the left fibration $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ X/}$ (Corollary 4.3.6.9), and is therefore also a left fibration (Remark 4.2.1.8). Since the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z)$ is a Kan complex (Proposition 4.6.6.4), the morphism $\pi '$ is a Kan fibration (Corollary 4.4.3.7). We will construct an auxiliary map of Kan complexes $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$ with the following properties:

$(a)$

The composition $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \lambda } Q \xrightarrow {\pi } \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(Y,Z)$ is equal to $\theta _{Y,Z}$.

$(b)$

The cubical diagram of Kan complexes

5.38
\begin{equation} \begin{gathered}\label{equation:diagram-Kan-slice-compatibility} \xymatrix@C =-10pt@R=40pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \ar [rr] \ar [dr]_{\rho \circ \overline{\theta } } \ar [dd] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \ar [dr]_{\lambda } \ar [dd]^(.6){\circ f} & \\ & \operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}}^{\mathrm{L}}( (Y,f), (Z,g) ) \ar [rr] \ar [dd] & & Q \ar [dd]^{\pi '} \\ \{ g\} \ar@ {=}[dr] \ar [rr] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [dr]_-{\theta _{X,Z}} & \\ & \{ g\} \ar [rr] & & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) } \end{gathered} \end{equation}

is commutative.

Suppose that such a map has been constructed. It follows from $(a)$ that $\lambda $ is a homotopy equivalence. Moreover, the front and back faces of the diagram (5.38) are pullback squares of simplicial sets. Since the vertical maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \quad \quad \pi ': Q \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) \]

are Kan fibrations, these faces are also homotopy pullback squares (Example 3.4.1.5). Since $\lambda $, $\theta _{X,Z}$, and the identity map $\operatorname{id}: \{ g\} \rightarrow \{ g\} $ are homotopy equivalences of Kan complexes, it follows from Corollary 3.4.1.10 that the map $\rho \circ \overline{\theta }$ is also a homotopy equivalence of Kan complexes. Since $\overline{\theta }$ is a homotopy equivalence, we conclude that $\rho $ is a homotopy equivalence as desired.

We now complete the proof by constructing the morphism $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$, so that $\theta _{Y,Z}( \sigma )$ is an $n$-simplex of the left-pinched morphism space $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(Y,Z)$ which we can identify with a simplicial functor $F_{\sigma }: \operatorname{Path}[ \{ y\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ such that $F_{\sigma }(y) = Y$ and $F_{\sigma } |_{ \operatorname{Path}[n]_{\bullet } }$ is the constant functor taking the value $Z$ (see Construction 4.6.7.3). We extend $F_{\sigma }$ to a simplicial functor $F_{\sigma }^{+}: \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as follows:

  • The functor $F_{\sigma }^{+}$ carries $x$ to the object $X \in \operatorname{\mathcal{C}}$.

  • For every element $i \in \{ y\} \star [n]$, the induced map of simplicial sets

    \[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ] }( x, i)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, F_{\sigma }(i) )_{\bullet } \]

    is given by the composition

    \begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ] }( x, i)_{\bullet } & \xrightarrow {u} & \operatorname{Hom}_{ \operatorname{Path}[ \{ y\} \star [n] ] }( y, i)_{\bullet } \\ & \xrightarrow { F_{\sigma } } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, F_{\sigma }(i) )_{\bullet } \\ & \xrightarrow { \circ f} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, F_{\sigma }(i) )_{\bullet }, \end{eqnarray*}

    where $u$ is induced by the map of partially ordered sets $\{ x\} \star \{ y\} \star [n] \rightarrow \{ y\} \star [n]$ which is the identity on $\{ y\} \star [n]$ and carries $x$ to $y$.

Then $F_{\sigma }^{+}$ determines a morphism of simplicial sets $\{ x \} \star \{ y\} \star \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carrying $\{ x\} \star \{ y\} $ to the edge $f$ and $\Delta ^{n}$ to the vertex $Z$, which we can identify with an $n$-simplex $\lambda (\sigma )$ of the Kan complex $Q$. The construction $\sigma \mapsto \lambda (\sigma )$ depends functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore induces a morphism of simplicial sets $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$ which is easily verified to satisfy conditions $(a)$ and $(b)$. $\square$