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Proposition 5.5.3.13. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. Then:

$(1)$

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

$(2)$

Let $(f,e): (C,x) \rightarrow (D,y)$ be an edge of the simplicial set $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ (see Remark 5.5.3.12). Then $(f,e)$ is $\pi $-cocartesian if and only if $e: \mathscr {F}(f)(C) \rightarrow y$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$.

Proof of Proposition 5.5.3.13. Suppose we are given integers $0 \leq i < n$ and a lifting problem

5.39
\begin{equation} \begin{gathered}\label{equation:strict-Grothendieck-construction-fibration} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar@ {^{(}->}[d] & \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \ar [d]^-{ \pi } \\ \Delta ^{n} \ar [r]^-{ \overline{\sigma }} \ar@ {-->}[ur]^{ \sigma } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \end{gathered} \end{equation}

Then $\overline{\sigma }$ can be identified with a composable chain of morphisms $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, and $\sigma _0$ determines a map of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \mathscr {F}(C_ n)$. Unwinding the definitions, we see that solutions to the lifting problem (5.39) can be identified with extensions of $\tau _0$ to an $n$-simplex $\tau : \Delta ^ n \rightarrow \mathscr {F}(C_ n)$. Since $\mathscr {F}(C_ n)$ is an $\infty $-category, such an extension always exists for $0 < i < n$. This proves that $\pi $ is an inner fibration of simplicial sets. In particular, the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ is an $\infty $-category.

Suppose now that $i = 0$, $n \geq 2$, and that the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )}$ corresponds to an edge $(f,e): (C,x) \rightarrow (D,y)$ of the simplicial set $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$. Then $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is the image of $e$ under the functor $\mathscr {F}(D) = \mathscr {F}(C_1) \rightarrow \mathscr {F}(C_ n)$. If $e$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$, then $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is an isomorphism in the $\infty $-category $\mathscr {F}(C_ n)$, so that $\tau _0$ can be extended to an $n$-simplex of $\mathscr {F}(C_ n)$ by virtue of Theorem 4.4.2.6. In particular, if $e$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$, then $(f,e)$ is a $\pi $-cocartesian morphism of the weighted nerve $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$.

Note that, for every object $(C,x)$ of the $\infty $-category $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ and every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, we can choose a morphism $(f,e): (C,x) \rightarrow (D,y)$ of the $\infty $-category $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$ where $e$ is an isomorphism in $\mathscr {F}(D)$: for example, we can take $y = \mathscr {F}(f)(x)$ and $e$ to be the identity morphism $\operatorname{id}_ y$. The preceding argument then shows that $(f,e)$ is $\pi $-cocartesian. This completes the proof of $(1)$.

To complete the proof of $(2)$, it will suffice to show that if $(f,e): (C,x) \rightarrow (D,y)$ is any $\pi $-cocartesian morphism of the $\infty $-category $\int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F}$, then $e$ is an isomorphism in the $\infty $-category $\mathscr {F}(D)$. To prove this, we can assume without loss of generality that $\operatorname{\mathcal{C}}= [1]$ is the linearly ordered set with two elements $C = 0$ and $D = 1$, in which case the desired result follows from Lemma 5.2.4.15 (see Example 5.5.3.8). $\square$