Kerodon

Example 4.1.2.4. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be its nerve. For every subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$, the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}')$ can be viewed as a subcategory of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is automatically an inner fibration, by virtue of Proposition 4.1.1.10). We will see in a moment that every subcategory of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ arises in this way (Corollary 4.1.2.11). In other words, when restricted to (the nerves of) ordinary categories, Definition 4.1.2.2 reduces to the classical notion of subcategory.