Corollary 5.4.7.14. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ denote the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then the projection maps
\[ \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/} \quad \quad \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0} \]
are interior fibrations of $(\infty ,2)$-categories.
Proof of Corollary 5.4.7.14.
We will show that the map of slice simplicial sets $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is an interior fibration; the analogous statement for coslice simplicial sets follows by a similar argument. We first observe that $\operatorname{\mathcal{C}}_{/f_0}$ is an $(\infty ,2)$-category (Corollary 5.4.3.4). Suppose we are given an integer $n \geq 2$ and a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d]^-{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}_{/f_0}. } \]
We wish to show that this lifting problem admits a solution provided that one of the following conditions is satisfied:
- $(a)$
The integer $i$ is equal to $0$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.
- $(b)$
The integer $i$ satisfies $0 < i < n$ and the restriction $\overline{\sigma }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}_{/f_0}$.
- $(c)$
The integer $i$ is equal to $n$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ n-1 < n \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.
In cases $(a)$ and $(c)$, this follows immediately from Proposition 5.4.3.8. In case $(b)$, it suffices (by virtue of Proposition 5.4.3.8) to verify that the composite map
\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Delta ^{n} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}} \]
is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. This follows from our hypothesis, since the projection map $\operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}}$ preserves thin $2$-simplices (Proposition 5.4.7.10).
$\square$