Proposition 5.4.6.14. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$. Assume that:
- $(1)$
Every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$.
- $(2)$
For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, there exists a thin $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ which belongs to $T$ and satisfies $d^{2}_2(\sigma ) = f$ and $d^{2}_0(\sigma ) = g$, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr] & & Z. } \]
- $(3)$
The collection $T$ has the inner exchange property (Definition 5.4.5.7).
- $(4)$
The collection $T$ has the four-out-of-five property (Definition 5.4.6.8).
Then every thin $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$.
Proof.
Let $\sigma $ be a thin $2$-simplex of $\operatorname{\mathcal{C}}$, whose $1$-skeleton we represent by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^-{h} & & Z. } \]
Applying assumption $(2)$, we can choose a thin $2$-simplex $\sigma '$ of $\operatorname{\mathcal{C}}$ which belongs to $T$ whose restriction to the $1$-skeleton of $\Delta ^2$ is represented by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^-{h'} & & Z. } \]
The edge $g$ determines a morphism of simplicial sets $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{E}}$ denote the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$. Since the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration (Proposition 5.4.3.1), it follows from Remark 5.4.2.4 and Example 5.4.2.2 that the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is an inner fibration; in particular, $\operatorname{\mathcal{E}}$ is an $\infty $-category. Moreover, we can identify the edges $f$, $h$, and $h'$ of $\operatorname{\mathcal{C}}$ with objects $\widetilde{Y}$, $\widetilde{Z}$, and $\widetilde{Z}'$ of $\operatorname{\mathcal{E}}$, and the $2$-simplices $\sigma $ and $\sigma '$ with morphisms $\widetilde{h}: \widetilde{Y} \rightarrow \widetilde{Z}$ and $\widetilde{h}': \widetilde{Y} \rightarrow \widetilde{Z}'$. Since $\sigma $ and $\sigma '$ are both thin, the morphisms $\widetilde{h}$ and $\widetilde{h}'$ are both $\pi $-cocartesian (Theorem 5.4.4.1). It follows that $\widetilde{h}$ and $\widetilde{h}'$ are isomorphic when viewed as objects of the $\infty $-category $\operatorname{\mathcal{E}}_{ \widetilde{Y} / }$ (see Remark 5.1.3.8). We can therefore choose a $2$-simplex $\rho $ of $\operatorname{\mathcal{E}}_{ \widetilde{Y} / }$ whose $1$-skeleton is given by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & \widetilde{h}' \ar [dr] & \\ \widetilde{h} \ar [rr]^-{ \operatorname{id}_{ \widetilde{h} }} \ar [ur] & & \widetilde{h}, } \]
which we can identify with a $4$-simplex $\tau : \Delta ^4 \rightarrow \operatorname{\mathcal{C}}$. For $0 \leq i < j < k \leq 4$, let $\tau _{kji}$ denote the $2$-simplex of $\operatorname{\mathcal{C}}$ given by $\tau |_{ \operatorname{N}_{\bullet }( \{ i < j < k \} )}$. By construction, the $2$-simplex $\tau _{310}$ is equal to $\sigma '$, and therefore belongs to $T$. Moreover, the $2$-simplices $\tau _{420}$, $\tau _{321}$, $\tau _{431}$, and $\tau _{432}$ are right-degenerate, and therefore belong to $T$ by virtue of assumption $(1)$. Since $T$ has the four-out-of-five-property, it follows that $\tau _{430}$ belongs to $T$. Applying the inner exchange property to the $3$-simplex $\tau |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3 < 4 \} )}$, we deduce that the $2$-simplex $\sigma = \tau _{410}$ also belongs to $T$, as desired.
$\square$