Proposition 5.4.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be interior fibrations of $(\infty ,2)$-categories. Then the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is also an interior fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Suppose we are given an integer $n \geq 2$ and a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{G \circ F} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{E}}. } \]
We wish to show that this lifting problem admits a solution if 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 $\operatorname{\mathcal{C}}$.
- $(b)$
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 $\operatorname{\mathcal{E}}$.
- $(c)$
The integer $i$ is equal to $n$ and $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ n-1 < n \} )}$ is a degenerate edge of $\operatorname{\mathcal{C}}$.
Since $G$ is an interior fibration, any of these hypotheses guarantee the existence of a solution to the associated lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{F \circ \sigma _0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^-{G} \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{E}}. } \]
It will therefore suffice to construct a solution to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{F} \\ \Delta ^{n} \ar [r]^-{\tau } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{D}}. } \]
In cases $(a)$ and $(c)$, our assumption that $F$ is an interior fibration immediately guarantees the existence of $\sigma $. In case $(b)$, it suffices to verify that the restriction $\tau |_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} )}$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$, which follows from Lemma 5.4.2.6. $\square$