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Proposition 5.5.2.8. Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category and let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration of simplicial sets. Then $\operatorname{\mathcal{C}}$ is also an $(\infty ,2)$-category.

Proof. We must verify that the simplicial set $\operatorname{\mathcal{C}}$ satisfies each of the axioms of Definition 5.5.1.3:

$(1)$

Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be edges of the simplicial set $\operatorname{\mathcal{C}}$; we wish to show that there exists a thin $2$-simplex $\Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ satisfying $d_2(\sigma ) = f$ and $d_0(\sigma ) = g$, as indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & Y \ar [dr]_{g} & \\ X \ar [ur]^{f} \ar [rr] & & Z. }$

We first invoke our assumption that $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category to choose a thin $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{D}}$ satisfying $d_2( \overline{\sigma } ) = q(f)$ and $d_0( \overline{\sigma } ) = q(g)$. Since $\overline{\sigma }$ is thin, our assumption that $q$ is an interior fibration guarantees that the lifting problem

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \ar [r]^-{ (g, \bullet , f) } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{q} \\ \Delta ^{2} \ar [r]^-{ \overline{\sigma } } \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{D}}}$

admits a solution. It follows from Lemma 5.5.2.6 that $\sigma$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(2)$

Let $\sigma$ be a degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. Then $q(\sigma )$ is a degenerate $2$-simplex of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$. Applying Lemma 5.5.2.6, we conclude that $\sigma$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(3)$

Let $n \geq 3$ and let $\tau _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1 < n\} ) }$ is left-degenerate; we wish to show that $\tau _0$ can be extended to an $n$-simplex $\tau$ of $\operatorname{\mathcal{C}}$. Let $\overline{\tau }_0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{D}}$ denote the composition $q \circ \tau _0$. Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category, we can extend $\overline{\tau }_0$ to an $n$-simplex $\overline{\tau }: \Delta ^{n} \rightarrow \operatorname{\mathcal{D}}$. To complete the proof, it will suffice to show that the lifting problem

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\tau _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{q} \\ \Delta ^{n} \ar [r]^-{ \overline{\tau } } \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{D}}}$

admits a solution. We conclude by observing that the edge $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0< 1\} ) }$ is degenerate and is therefore $q$-cocartesian by virtue of our assumption that $q$ is an interior fibration.

$(4)$

Let $n \geq 3$ and let $\tau _0: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets with the property that the $2$-simplex $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0< n-1 < n\} ) }$ is right-degenerate; we wish to show that $\tau _0$ can be extended to an $n$-simplex $\tau$ of $\operatorname{\mathcal{C}}$. This follows by the argument given above, applied to the opposite interior fibration $q^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$.

$\square$