Lemma 5.3.6.11 (02UA)

Lemma 5.3.6.11. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a cocartesian fibration of simplicial sets, let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of simplicial sets, and let $e$ be an edge of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ which satisfies condition $(\ast )$ of Proposition 5.3.6.6. Then $e$ is $V'$-cartesian, where $V': \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ is given by postcomposition with $V$.

Proof. Let $n \geq 2$ and suppose we are given a lifting problem

5.30

\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [r]^-{ \sigma _0 } \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \ar [d]^{V'} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}}), } \end{gathered} \end{equation}

where $\sigma _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^{n}_{n}$ to $e$; we wish to show that this lifting problem admits a solution. Replacing $U$ with the projection map $\Delta ^{n} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ n$, we can assume without loss of generality that $\operatorname{\mathcal{B}}= \Delta ^ n$ is a standard simplex, so that $\overline{\sigma }$ corresponds to a morphism of simplicial sets $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$. Invoking the universal property of the simplicial sets $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}(\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{E}})$ (Proposition 4.5.9.5), we can rewrite (5.30) as a lifting problem

5.31

\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image-3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [d] \ar [r]^-{F_0} & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{F} } \ar@ {-->}[ur]^{F} & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}

Note that since the edge $e$ satisfies condition $(\ast )$, the morphism $F_0$ satisfies the following condition:

If $u$ is a $U$-cocartesian edge of $\operatorname{\mathcal{C}}$ lying over the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Delta ^ n$, then $F_0(u)$ is a $V$-cartesian edge of $\operatorname{\mathcal{D}}$.

Using Corollary 5.3.4.9, we can choose a diagram of $\infty $-categories $\mathscr {F}: [n] \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{C}}$. Set $\operatorname{\mathcal{C}}' = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\operatorname{\mathcal{C}}'_0 = \Lambda ^{n}_{n} \times _{ \Delta ^ n } \operatorname{\mathcal{C}}'$, so that $\lambda $ restricts to a map $\lambda _0: \operatorname{\mathcal{C}}'_0 \rightarrow \operatorname{\mathcal{C}}_0$. We then have a commutative diagram of $\infty $-categories

5.32

\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image-2} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ \lambda } \ar [d] & \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}', \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}}) \ar [r]^-{\circ \lambda } & \operatorname{Fun}_{ / \operatorname{\mathcal{E}}}( \operatorname{\mathcal{C}}'_0, \operatorname{\mathcal{D}}). } \end{gathered} \end{equation}

Since $V$ is an isofibration (Proposition 5.1.4.9), the vertical maps in this diagram are isofibrations (Proposition 4.5.5.14). Since $\lambda $ and $\lambda _0$ are categorical equivalences of simplicial sets (Lemma 5.3.6.4), the horizontal maps are equivalences of $\infty $-categories. Applying Corollary 4.5.2.32, we deduce that the upper horizontal map in the diagram (5.32) restricts to an equivalence from each fiber of the left vertical map to the corresponding fiber of the right vertical map. Consequently, we can replace (5.31) with the lifting problem

5.33

\begin{equation} \begin{gathered}\label{equation:cartesian-in-direct-image-4} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}'_0 \ar [d] \ar [r]^-{F_0 \circ \lambda _0} & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r]^-{\overline{F} \circ \lambda } \ar@ {-->}[ur] & \operatorname{\mathcal{E}}. } \end{gathered} \end{equation}

Using Remark 5.3.2.12, we obtain a pushout square

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \times \mathscr {F}(0) \ar [r]^-{G_0} \ar [d] & \operatorname{\mathcal{C}}'_0 \ar [d] \\ \Delta ^ n \times \mathscr {F}(0) \ar [r]^-{G} & \operatorname{\mathcal{C}}'. } \]

Let us identify $F_0 \circ \lambda _0 \circ G_0$ with a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{n} \rightarrow \operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{D}})$, and $\overline{F} \circ \lambda \circ G$ with an $n$-simplex $\overline{\tau }$ of $\operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}})$, so that we can rewrite (5.33) again as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n} \ar [d] \ar [r]^-{ \tau _0 } & \operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}}) \ar [d]^{V''} \\ \Delta ^{n} \ar [r]^-{\overline{\tau }} \ar@ {-->}[ur]^{\tau } & \operatorname{Fun}(\mathscr {F}(0), \operatorname{\mathcal{D}}). } \]

To show that this lifting problem admits a solution, it will suffice to show that $\tau _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} )$ of $\Lambda ^{n}_{n}$ to a $V''$-cocartesian edge of the simplicial set $\operatorname{Fun}( \mathscr {F}(0), \operatorname{\mathcal{E}})$. Since $\lambda $ is a scaffold, this follows by combining $(\ast ')$ with the criterion of Theorem 5.2.1.1. $\square$