Kerodon

Proof. We will prove the first assertion; the proof of the second is similar. Fix a simplicial functor $G: \operatorname{\mathcal{D}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ and set $Y = G(Y_0)$. We wish to show that there is a unique simplicial functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$ for which the composition

is equal to $G$. For each object $D \in \operatorname{\mathcal{D}}$, the simplicial functor $G$ induces a morphism of simplicial sets

which we can identify with a vertex $f$ of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), Y)_{\bullet }$. The simplicial functor $F$ is then given on objects by the formula $F(D) = (G(D), f)$, and is determined on morphisms by the requirement that the composition

coincides with the map of simplicial sets determined by the simplicial functor $G$. $\square$

Proof of Theorem 5.5.2.21 from Proposition 5.5.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which satisfies hypothesis $(\ast )$ of Theorem 5.5.2.21. Applying Proposition 5.5.2.22 in the case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{X/}$, we conclude that the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ is fully faithful. Since $c'$ is bijective on vertices, it is also essentially surjective, and is therefore an equivalence of $\infty $-categories by virtue of Theorem 4.6.2.21. $\square$

Proof of Proposition 5.5.2.22. 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 5.5.2.22. 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

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

By virtue of Proposition 4.6.5.10, 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 4.6.8.3 supplies comparison maps

which are homotopy equivalences of Kan complexes by virtue of Theorem 4.6.8.5. 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 4.3.6.13), and therefore restricts to a trivial Kan fibration

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

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 4.3.6.12), 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 4.6.5.5), the morphism $\pi '$ is a Kan fibration (Corollary 4.4.3.8). 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

5.56

\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 (5.56) are pullback squares of simplicial sets. Since the vertical maps

are Kan fibrations, these faces are also homotopy pullback squares (Example 3.4.1.3). Since $\lambda $, $\theta _{X,Z}$, and the identity map $\operatorname{id}: \{ g\} \rightarrow \{ g\} $ are homotopy equivalences of Kan complexes, it follows from Corollary 3.4.1.12 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 4.6.8.3). 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 $(b)$. $\square$