Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 5.3.7.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration of $(\infty ,2)$-categories (Definition 5.3.2.1). Then:

$(1)$

The morphism $F$ is a functor of $(\infty ,2)$-categories: that is, it carries thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$, and therefore induces a functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$.

$(2)$

The diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Pith}(\operatorname{\mathcal{C}}) \ar [r] \ar [d]^-{ \operatorname{Pith}(F) } \ar [r] & \operatorname{\mathcal{C}}\ar [d]^-{F} \\ \operatorname{Pith}(\operatorname{\mathcal{D}}) \ar [r] & \operatorname{\mathcal{D}}} \]

is a pullback square.

$(3)$

The functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$ is an inner fibration of $\infty $-categories.

Proof. We will prove assertion $(1)$ by showing that $F$ satisfies the criterion of Proposition 5.3.7.8. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{D}}$ is an $(\infty ,2)$-category, we can choose a thin $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{D}}$ satisfying $d_0( \overline{\sigma } ) = F(g)$ and $d_2( \overline{\sigma } ) = F(f)$, which we depict as a diagram

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

Since $F$ is an interior fibration, the lifting problem

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

admits a solution. Then $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (Lemma 5.3.2.6) for which the image $\overline{\sigma } = F(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.

We now prove $(2)$. Let $\tau $ be an $m$-simplex of the simplicial set $\operatorname{\mathcal{C}}$, and suppose that $F(\tau )$ belongs to the pith $\operatorname{Pith}(\operatorname{\mathcal{D}})$. We wish to show that $\tau $ belongs to $\operatorname{Pith}(\operatorname{\mathcal{C}})$: that is, that it carries each $2$-simplex of $\Delta ^ m$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. This follows immediately from Lemma 5.3.2.6, since the composite map

\[ \Delta ^2 \rightarrow \Delta ^ m \xrightarrow {\tau } \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{D}}$.

Combining $(2)$ with Remark 5.3.2.4, we conclude that the functor $\operatorname{Pith}(F): \operatorname{Pith}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Pith}(\operatorname{\mathcal{D}})$ is an interior fibration. Since $\operatorname{Pith}(\operatorname{\mathcal{D}})$ is an $\infty $-category (Proposition 5.3.5.6), it follows that $\operatorname{Pith}(F)$ is an inner fibration (Example 5.3.2.2). $\square$