Proposition 7.2.3.12 (02P6)

Proposition 7.2.3.12. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d] \ar [r]^-{F} & \operatorname{\mathcal{C}}\ar [d]^-{\pi } \\ \operatorname{\mathcal{D}}' \ar [r]^-{\overline{F}} & \operatorname{\mathcal{D}}. } \]

If $\pi $ is a cocartesian fibration and $\overline{F}$ is right cofinal, then $F$ is right cofinal.

Proof. By virtue of Corollary 7.2.1.15, it will suffice to prove Proposition 7.2.3.12 in the special case where $\overline{F}$ is right anodyne. Let $S$ be the collection of all morphisms of simplicial sets $\overline{F}: \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}$ having the property that, for every cocartesian fibration $\pi : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the induced map $F: \operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is right anodyne. We wish to show show that every right anodyne morphism belongs to $S$. It follows immediately from the definitions that $S$ is weakly saturated, in the sense of Definition 1.5.4.12. It will therefore suffice to show that $S$ contains every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 < i \leq n$. In other words, we are reduced to proving Proposition 7.2.3.12 in the special case where $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex and $\overline{F}$ is the inclusion of the horn $\Lambda ^{n}_{i} \subseteq \Delta ^ n$.

Applying Corollary 5.3.4.9, we deduce that there exists a diagram of $\infty $-categories $\mathscr {G}: [n] \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \rightarrow \operatorname{\mathcal{C}}$ for the cocartesian fibration $\pi $. We then have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \times _{\Delta ^ n} \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G}) \ar [r] \ar [d]^{F'} & \Lambda ^{n}_{i} \times _{\Delta ^ n} \operatorname{\mathcal{C}}\ar [d]^{F} \\ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G}) \ar [r]^-{\lambda } & \operatorname{\mathcal{C}}, } \]

where $F'$ is right anodyne (Lemma 7.2.3.10) and therefore right cofinal (Proposition 7.2.1.3). Lemma 5.3.6.4 guarantees that horizontal maps are categorical equivalences, so that $F$ is also right cofinal (Corollary 7.2.1.22). $\square$