Kerodon

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

Proposition 5.3.3.15. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. Then:

$(1)$

The projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a cocartesian fibration of simplicial sets.

$(2)$

Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Remark 5.3.3.4). Then $(f,e)$ is $U$-cocartesian if and only if $e: \mathscr {F}(f)(x) \rightarrow y$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$.

Proof. By virtue of Proposition 5.1.4.7 and Remark 5.3.3.7, we may assume without loss of generality that $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ for some nonnegative integer $n$. We proceed by induction on $n$. If $n = 0$, then $U$ can be identified with the projection map $\mathscr {F}(0) \rightarrow \Delta ^0$ (Example 5.3.3.2), so that assertions $(1)$ and $(2)$ follow from Examples 5.1.4.3 and 5.1.1.4, respectively. Let us therefore assume that $n > 0$, so that $\operatorname{\mathcal{C}}$ can be identified with the cone $\operatorname{\mathcal{C}}_{0}^{\triangleright }$ for $\operatorname{\mathcal{C}}_0 = [n-1]$. Set $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$. It follows from our inductive hypothesis that the projection map $U_0: \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is a cocartesian fibration of $\infty $-categories, and that a morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0)$ is $U_0$-cocartesian if and only if it satisfies the criterion described in $(2)$. It follows that the functor $\operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \mathscr {F}(n)$ described in Example 5.3.3.12 carries $U_0$-cocartesian morphisms to isomorphisms in the $\infty $-category $\mathscr {F}(n)$. Unwinding the definitions, we can identify $U$ with the map of relative joins

\[ \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \star _{\mathscr {F}(n)} \mathscr {F}(n) \rightarrow \Delta ^{n-1} \star _{ \Delta ^0} \Delta ^{0}. \]

Assertion $(1)$ now follows from Lemma 5.2.3.17. To prove $(2)$, we can assume without loss of generality that $n=1$, in which case the desired result follows from Example 5.2.3.18 (see Example 5.3.3.14). $\square$