Kerodon

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 cocartesian fibration $\alpha _0: \mathscr {F}(0) \rightarrow \mathscr {G}(0)$ (Example 5.3.3.2), so that assertions $(1)$ and $(2)$ are immediate. 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 $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ and $\mathscr {G}_0 = \mathscr {G}|_{ \operatorname{\mathcal{C}}_0 }$. It follows from our inductive hypothesis that $U$ restricts to a cocartesian fibration $U_0: \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {G}_0}(\operatorname{\mathcal{C}}_0)$ is a cocartesian fibration of $\infty $-categories, and that an edge 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 $\alpha _{n}$-cocartesian morphisms of the $\infty $-category $\mathscr {F}(n)$. Unwinding the definitions, we can identify $U$ with the map of relative joins

Assertions $(1)$ and $(2)$ now follow from Lemma 5.2.3.17. $\square$

Proof. Apply Proposition 5.3.3.15 in the special case where $\mathscr {G}$ is the constant diagram taking the value $\Delta ^0$. $\square$

Proof. Combine Propositions 5.1.4.14 and 5.3.3.15. $\square$

Proof. Apply Corollary 5.3.3.17 in the special case where $\mathscr {G}$ is the constant diagram taking the value $\Delta ^0$. $\square$

Proof. Assume first that $\alpha $ is a levelwise categorical equivalence. To prove that $T$ is a categorical equivalence of simplicial sets, it will suffice to show that for every simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map $T_{\sigma }: \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ is a categorical equivalence of simplicial sets (Corollary 4.5.7.3). Using Remark 5.3.3.7, we can reduce to the special case where $\operatorname{\mathcal{C}}$ is the linearly ordered $[n] = \{ 0 < 1 < \cdots < n \} $ for some $n \geq 0$. We now proceed by induction on $n$. If $n = 0$, the result is immediate from Example 5.3.3.2. The inductive step follows by combining Example 5.3.3.12 with Corollary 5.2.4.7.

We now prove the converse. Using Proposition 4.1.3.2, we can choose a commutative diagram

in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, where the vertical maps are levelwise categorical equivalences and the simplicial sets $\mathscr {G}(C)$ and $\mathscr {G}'(C)$ are $\infty $-categories for each $C \in \operatorname{\mathcal{C}}$. Using the first part of the proof, we can replace $\alpha $ by $\beta $ and thereby reduce to the special case where $\mathscr {F}$ and $\mathscr {F}'$ are diagrams of $\infty $-categories. In this case, the projection maps $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \leftarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}})$ are cocartesian fibrations of $\infty $-categories (Corollary 5.3.3.16). It then follows from Theorem 5.1.6.1 (together with Example 5.3.3.8) that if $T$ is an equivalence of $\infty $-categories, then $\alpha $ is a levelwise categorical equivalence. $\square$

Proof of Proposition 5.3.3.23. The construction $\operatorname{\mathcal{E}}\mapsto T_{\operatorname{\mathcal{E}}}$ carries colimits in the category $(\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to limits in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$. We can therefore assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex, so that the morphism $U$ determines a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we see that the codomain of $T_{\operatorname{\mathcal{E}}}$ can be identified with the set of tuples $\tau = (\tau _0, \tau _1, \cdots , \tau _ n)$, where $\tau _ i: \Delta ^{i} \rightarrow \mathscr {F}(C_ i)$ are simplices for which the diagram

is commutative. Let us regard $\tau $ as fixed; we wish to prove that there is a unique natural transformation $\alpha : \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ) = \tau $.

Let $D$ be an object of $\operatorname{\mathcal{C}}$ and let $m \geq 0$ be an integer. Then $m$-simplices of the simplicial set $\mathscr {G}_{\operatorname{\mathcal{E}}}(D) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$ can be identified with pairs $(f, g)$, where $g: [m] \rightarrow [n]$ is a nondecreasing function and $f: C_{ g(m) } \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$. Let $\alpha _{D}(f,g)$ denote the $m$-simplex of $\mathscr {F}(D)$ given by the composition

The construction $(f,g) \mapsto \alpha _{D}(f,g)$ determines a morphism of simplicial sets $\alpha _{D}: \mathscr {G}_{\operatorname{\mathcal{E}}}(D) \rightarrow \mathscr {F}(D)$. The assignment $D \mapsto \alpha _{D}$ determines a natural transformation of functors $\alpha : \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ) = \tau $. This proves existence.

We now prove uniqueness. Suppose we are given another natural transformation $\alpha ': \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ') = \tau $; we wish to show that $\alpha = \alpha '$. Fix an object $D \in \operatorname{\mathcal{C}}$ and an $m$-simplex of the simplicial set $\mathscr {G}_{\operatorname{\mathcal{E}}}(D)$, which we identify with a pair $(f,g)$ as above. We wish to verify that $\alpha _{D}(f,g)$ and $\alpha '_{D}(f,g)$ coincide (as $m$-simplices of the simplicial set $\mathscr {F}(D)$). Set $n' = g(m)$, so that the function $g$ factors as a composition $[m] \xrightarrow {g} [n'] \xrightarrow {\iota } [n]$, where $\iota : [n'] \hookrightarrow [n]$ is the inclusion map. Since $\alpha _ D$ and $\alpha '_{D}$ are morphisms of simplicial sets, it will suffice to prove that $\alpha _{D}( f, \iota )$ and $\alpha '_{D}(f, \iota )$ coincide (as $n'$-simplices of the simplicial set $\mathscr {F}(D)$). Since both $\alpha _{D}$ and $\alpha '_{D}$ are natural in $D$, we may assume without loss of generality that $D = C_{n'}$ and that $f$ is the identity morphism. In this case, the identities $T_{\operatorname{\mathcal{E}}}( \alpha ) = \tau = T_{\operatorname{\mathcal{E}}}( \alpha ' )$ give $\alpha _{D}(f,\iota ) = \tau _{n'} = \alpha '_{D}(f,\iota )$. $\square$