Kerodon

Proof of Proposition 5.4.5.10. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $\sigma : \Delta ^{3} \rightarrow \operatorname{\mathcal{C}}$ be a $3$-simplex of $\operatorname{\mathcal{C}}$ and let $C = \sigma (3) \in \operatorname{\mathcal{C}}$ be the image of the final vertex. Let us regard the face $\sigma _{210} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )}$ as a morphism of simplicial sets from $\Delta ^2$ to $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ denote the pullback $\Delta ^{2} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the projection map $\operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration (Proposition 5.4.3.1). If $\sigma _{210}$ is thin, then the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ is also an interior fibration (Remark 5.4.2.4); since $\Delta ^2$ is an $\infty $-category, it is an inner fibration (Example 5.4.2.2). Unwinding the definitions, we can identify $\sigma $ with a $2$-simplex of $\operatorname{\mathcal{E}}$ lying over the unique nondegenerate $2$-simplex of $\Delta ^2$, which we display as a diagram

If $\sigma _{321} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} )}$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, then the “easy direction” of Theorem 5.4.4.1 guarantees that $g$ is $\pi $-cartesian. It follows that $f$ is $\pi $-cartesian if and only if $h$ is $\pi $-cartesian (Corollary 5.1.2.4). Equivalently, $f$ is locally $\pi $-cartesian if and only if $h$ is locally $\pi $-cartesian (see Remark 5.1.3.4). Applying the “hard direction” of Theorem 5.4.4.1, we conclude that the $2$-simplex $\sigma _{310} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3\} ) }$ is thin if and only if the $2$-simplex $\sigma _{320} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < 3 \} )}$ is thin. $\square$

Proof of Proposition 5.4.5.6. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Suppose we are given integers $0 < i < n$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$. If $n = 2$, then condition $(1)$ of Definition 5.4.1.1 guarantees that we can extend $\sigma _0$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$, which then belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$ by virtue of Remark 5.4.5.2. We may therefore assume that $n \geq 3$. In this case, we observe that the composite map

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, so that we can extend $\sigma _0$ to an $n$-simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that $\sigma $ carries each $2$-simplex of $\Delta ^ n$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. If $n \geq 4$, this is automatic (since every $2$-simplex of $\Delta ^ n$ is contained in the horn $\Lambda ^{n}_{i}$). In the case $n=3$, it follows from our assumption that the collection of thin $2$-simplices of $\operatorname{\mathcal{C}}$ has the inner exchange property (Proposition 5.4.5.10). $\square$

Proof. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$, which we view as a morphism of simplicial sets $\tau : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ whose restriction to the face $\Delta ^{n} \subseteq \Delta ^{n+1}$ equal to the constant map $\Delta ^{n} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$. Then $\sigma $ belongs to the simplicial subset $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \subseteq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ if and only if, for every $2$-simplex $\rho : \Delta ^2 \rightarrow \Delta ^{n+1}$, the composition $\tau \circ \rho $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Note that this condition is automatically satisfied if $\rho $ is degenerate, or takes values in the subset $\Delta ^{n} \subseteq \Delta ^{n+1}$ (since every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ is thin). Consequently, it suffices to verify this condition in the case where $\rho $ is the right cone of a map $\rho _0: \Delta ^1 \rightarrow \Delta ^{n}$. In this case, $\tau \circ \rho $ is thin if and only if the edge $\Delta ^{1} \xrightarrow {\rho _0} \Delta ^{n} \xrightarrow {\sigma } \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)$ (Theorem 5.4.4.1). Allowing $\tau _0$ to vary, we obtain the identification $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y) \simeq \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}( X, Y)^{\simeq }$; the proof of the analogous statement for left-pinched morphism spaces is similar. $\square$

Proof. We will prove $(1)$; the proof of $(2)$ is similar. It follows from Remark 5.4.2.4 that $\pi $ is an interior fibration. Since $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Proposition 5.4.5.6), it is an inner fibration of $\infty $-categories (Example 5.4.2.2). Let us say that a morphism $u$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is special if, for every vertex $z \in K$, the composite map

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Let $\overline{\pi }: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. It follows from Corollary 5.4.4.2 that every special morphism of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is $\overline{\pi }$-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/f}$, and therefore also $\pi $-cartesian (Remark 5.1.1.12). Conversely, any $\pi $-cartesian morphism of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ is locally $\overline{\pi }$-cartesian when viewed as a morphism of $\operatorname{\mathcal{C}}_{/f}$, and therefore special (again by Corollary 5.4.4.2). To complete the proof, it will suffice to show that if $Y$ is an object of $\operatorname{\mathcal{C}}_{/f}$, then any morphism $\overline{u}: \overline{X} \rightarrow q( \overline{Y} )$ in $\operatorname{Pith}( \operatorname{\mathcal{C}})$ can be lifted to a special morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/f} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$, which follows from Proposition 5.4.3.9. $\square$