Lemma 7.3.4.5. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty $-categories, let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a simplicial subset which contains every vertex of $\operatorname{\mathcal{C}}$, and set $\operatorname{\mathcal{K}}_0 = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}$. Suppose we are given a lifting problem
7.14
\begin{equation} \begin{gathered}\label{equation:fiberwise-relative-Kan-general} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\coprod _{\operatorname{\mathcal{K}}_0} (\operatorname{\mathcal{K}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0) \ar [r]^-{F_0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{E}}} \end{gathered} \end{equation}
which satisfies the following condition:
- $(\ast )$
Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex which is not contained in $\operatorname{\mathcal{C}}_0$ and set $C = \sigma (0)$. Then the composite map
\[ \operatorname{\mathcal{K}}_ C^{\triangleright } \simeq \operatorname{\mathcal{K}}_{C} \star _{ \{ C\} } \{ C\} \hookrightarrow \operatorname{\mathcal{K}}_0 \star _{ \operatorname{\mathcal{C}}_0 } \operatorname{\mathcal{C}}_0 \xrightarrow {F_0} \operatorname{\mathcal{D}} \]
is a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$.
Then the lifting problem (7.14) admits a solution.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category (working one simplex at a time, we could even assume that $\operatorname{\mathcal{C}}= \Delta ^ m$ is a standard simplex and that $\operatorname{\mathcal{C}}_0 = \operatorname{\partial \Delta }^ m$ is its boundary). Set $\overline{\operatorname{\mathcal{K}}} = \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, so that $\delta $ extends to a map
\[ \overline{\delta }: \overline{\operatorname{\mathcal{K}}} = \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\simeq \Delta ^1 \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}. \]
Since $\delta $ is a cocartesian fibration, Lemma 5.2.3.17 guarantees that $\overline{\delta }$ is also a cocartesian fibration. Applying Proposition 5.3.6.6, we obtain a commutative diagram of $\infty $-categories
7.15
\begin{equation} \begin{gathered}\label{equation:diagram-of-Res} \xymatrix@R =50pt@C=50pt{ \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}} / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}}) \ar [rr] \ar [dd]^{U \circ } \ar [dr]^{T} & & \operatorname{Res}_{\operatorname{\mathcal{K}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \ar [dd]^{U \circ } \ar [dl] \\ & \operatorname{\mathcal{C}}& \\ \operatorname{Res}_{ \overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) \ar [rr] \ar [ur] & & \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}), \ar [ul] } \end{gathered} \end{equation}
where the diagonal arrows are cartesian fibrations and the morphisms on the outside of the diagram preserve cartesian morphisms. Applying Proposition 5.1.4.20, we see that the induced map
\[ T': \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \times _{ \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}) } \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) \rightarrow \operatorname{\mathcal{C}} \]
is also a cartesian fibration, and that the outer square of the diagram (7.15) determines a functor
\[ V: \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}}) \rightarrow \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \times _{ \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}) } \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) \rightarrow \operatorname{\mathcal{C}} \]
which carries $T$-cartesian morphisms to $T'$-cartesian morphisms. Moreover, the functor $V$ is an isofibration (Proposition 4.5.9.17).
Unwinding the definitions, we can rewrite (7.14) as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r]^-{ G_0} \ar [d] & \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}}) \ar [d]^{V} & \\ \operatorname{\mathcal{C}}\ar [r] \ar@ {-->}[ur] & \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}\times \operatorname{\mathcal{K}}) \times _{ \operatorname{Res}_{ \operatorname{\mathcal{K}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \operatorname{\mathcal{K}}) } \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}\times \overline{\operatorname{\mathcal{K}}}) . } \]
By virtue of Corollary 7.1.6.6, to show that this lifting problem admits a solution, it will suffice to verify the following:
- $(\ast ')$
Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be an $n$-simplex which is not contained in $\operatorname{\mathcal{C}}_0$ and set $C = \sigma (0)$. Then $G_0(C)$ is a $V$-initial object of the $\infty $-category $ \operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}})$.
Unwinding the definitions, we see that the functor $T^{-1} \{ C\} \rightarrow T'^{-1} \{ C\} $ induced by $V$ can be identified with the restriction map
\[ V_ C: \operatorname{Fun}( \operatorname{\mathcal{K}}_{C}^{\triangleright }, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{K}}_ C, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \operatorname{\mathcal{K}}_ C, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{K}}_ C^{\triangleright }, \operatorname{\mathcal{E}}). \]
Combining assumption $(\ast )$ with Proposition 7.1.6.3, we see that $G_0( C )$ is a $V_ C$-initial object of the $\infty $-category $\operatorname{Fun}( K_{C}^{\triangleright }, \operatorname{\mathcal{D}})$. Proposition 7.1.4.19 then guarantees that $G_0(C)$ is also $V$-initial when regarded as an object of the $\infty $-category $\operatorname{Res}_{\overline{\operatorname{\mathcal{K}}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{D}}\times \overline{\operatorname{\mathcal{K}}})$.
$\square$