Kerodon

Example 4.4.3.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, and let $\operatorname{\mathcal{C}}^{\simeq }$ denote its core (in the sense of Construction 1.3.5.4). Then the core of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Construction 4.4.3.1) can be identified with the nerve of $\operatorname{\mathcal{C}}^{\simeq }$. That is, we have a canonical isomorphism $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\simeq } \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\simeq } )$.