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Example Let $n$ be an integer. Then every $(n,1)$-category is locally $(n-1)$-truncated (Corollary In particular:

  • If $Q$ is a partially ordered set, then the nerve $\operatorname{N}_{\bullet }(Q)$ is locally $(-1)$-truncated $\infty $-category (Proposition

  • If $\operatorname{\mathcal{C}}$ is an ordinary category, then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a locally $0$-truncated $\infty $-category (Example

  • If $\operatorname{\mathcal{C}}$ is a $2$-category in which every $2$-morphism is an isomorphism, then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a locally $1$-truncated $\infty $-category (Example