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Proposition 5.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the projection maps

\[ \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}} \]

are interior fibrations.

Proof of Proposition 5.3.3.1. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will show that the projection map $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration; the analogous assertion for the coslice simplicial set $\operatorname{\mathcal{C}}_{f/}$ follows by a similar argument. Let $m \geq 2$ and suppose that we are given a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d] \\ \Delta ^{m} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur] & \operatorname{\mathcal{C}}. } \]

We wish to show that a solution exists under any of the following additional assumptions:

$(a)$

The integer $i$ is equal to zero and the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

$(b)$

The integer $i$ satisfies $0 < i < m$ and the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Delta ^{m} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(c)$

The integer $i$ is equal to $m$ and the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ m-1 < m \} ) }$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

In cases $(a)$ and $(b)$, this follows immediately from Proposition 5.3.3.8. In case $(c)$, we observe that for every vertex $x \in K$, the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \star \{ x\} \hookrightarrow \Lambda ^{m}_{i} \star K \xrightarrow { \sigma _0} \operatorname{\mathcal{C}} \]

is a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, this degenerate $2$-simplex is thin, so that existence of the desired extension again follows from Proposition 5.3.3.8. $\square$