Kerodon

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

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). $\square$

Proof. By virtue of Corollary 4.5.4.3, it will suffice to show that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map

is a categorical equivalence of simplicial sets. Let us identify $\sigma $ with a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, so that the fiber product $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }( \operatorname{Set_{\Delta }})$ is the mapping simplex $M( \mathscr {F}(C_0) \rightarrow \cdots \rightarrow \mathscr {F}(C_ n) )$ introduced in Construction 5.2.6.3. Let us abuse notation by identifying each $\mathscr {F}(C_ i)$ with the fiber of the projection map

over the $i$th vertex of $\Delta ^ n$. It follows from Proposition 5.5.3.13 that $\pi $ is a cocartesian fibration and that $F_{\sigma }$ is a scaffold of $\pi $, in the sense of Definition 5.2.6.12. Applying Proposition 5.2.6.19, we conclude that $F_{\sigma }$ is a categorical equivalence of simplicial sets. $\square$