Kerodon

Proof. By definition, the morphism $\theta $ is a trivial Kan fibration if and only if every lifting problem

admits a solution. For $m \leq n$ this condition is vacuous, and for $m > n$ it is a reformulation of the criterion of Remark 4.7.6.11. $\square$

Proof. By virtue of Corollary 4.7.6.16, it will suffice to show that $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is an $\infty $-category. We proceed as in the proof of Proposition 3.5.3.23. Fix integers $0 < i < m$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$; we wish to show that $\sigma _0$ can be extended to an $m$-simplex of $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$. Using Remark 3.5.3.21, we can identify $\sigma _0$ with a morphism of simplicial sets $f_0: \operatorname{sk}_{n}( \Lambda ^{m}_{i} ) \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $f_0$ can be extended to the $n$-skeleton of $\Delta ^{m}$. If $n < m-1$, then $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \operatorname{sk}_{n}( \Delta ^{m} )$ and there is nothing to prove. We may therefore assume that $n \geq m-1$, so that $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \Lambda ^{m}_{i}$. In this case, our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category guarantees that $f_0$ can be extended $\Delta ^ m$ (and therefore also the the $n$-skeleton of $\Delta ^ m$). $\square$

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.7.6.15, the implication $(2) \Rightarrow (3)$ from Corollaries 4.7.6.17 and 4.5.4.12 the implication $(3) \Rightarrow (4)$ is trivial, and the implication $(4) \Rightarrow (1)$ follows from Corollary 4.7.6.16 and Remark 4.7.6.4. $\square$

Proof. It follows from Corollary 4.7.6.17 that $\operatorname{cosk}_{n}(\operatorname{\mathcal{C}})$ is a locally $(n-2)$-truncated $\infty $-category. It will therefore suffice to show that, for every locally $(n-2)$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $\theta : \operatorname{Fun}( \operatorname{cosk}_{n}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. By virtue of Corollary 4.7.6.18, we may assume without loss of generality that $\operatorname{\mathcal{E}}$ is $n$-coskeletal. In this case, $\theta $ is an isomorphism of simplicial sets (Proposition 3.5.3.17). $\square$

Proof. Assume that condition $(1)$ is satisfied. By virtue of Proposition 4.7.6.19, we may assume without loss of generality that $F$ is the canonical map from $\operatorname{\mathcal{C}}$ to its $(n+2)$-coskeleton $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$. In this case, assertion $(2)$ follows from Proposition 3.5.3.23 and Example 3.5.7.24. The reverse implication follows by a similar argument. $\square$

Proof of Corollary 4.7.6.24. Our assumption on $F$ guarantees that the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) )$ is $n$-truncated. It will therefore suffice to show that $F_{X,Y}$ is $(n+1)$-connective, or equivalently that the map of left-pinched morphism spaces $\theta : \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{D}}}( F(X), F(Y))$ is $(n+1)$-connective (Proposition 4.6.5.10). By virtue of Proposition 4.7.6.19, we may assume without loss of generality that $F$ exhibits $\operatorname{\mathcal{D}}$ as an $(n+2)$-coskeleton of $\operatorname{\mathcal{C}}$ and is therefore bijective on simplices of dimension $\leq n+2$. In this case, $\theta $ is bijective on simplices of dimension $\leq n+1$, so the desired result follows from Corollary 3.5.2.2. $\square$

Proof. By virtue of Corollary 4.7.6.24, we can replace $\operatorname{\mathcal{C}}$ by $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and thereby reduce to the case where the $\infty $-category $\operatorname{\mathcal{C}}$ is $(n+2)$-coskeletal, and therefore locally $n$-truncated. In this case, condition $(1)$ is satisfied if and only if $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. The equivalence $(1) \Leftrightarrow (2)$ now follows from the criterion of Remark 4.7.6.11. $\square$