Kerodon

Example 9.2.2.3. Let $\operatorname{\mathcal{C}}$ be a category which admits small filtered colimits. Then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category which admits small filtered colimits (Corollary 9.1.9.13). Moreover, an object $C \in \operatorname{\mathcal{C}}$ is compact (in the sense of Definition 9.2.0.2) if and only if it is compact when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 9.2.2.1). This follows from Proposition 9.1.9.10, since the inclusion functor $\operatorname{N}_{\bullet }(\operatorname{Set}) \hookrightarrow \operatorname{\mathcal{S}}$ is finitary (see Variant 9.1.6.4).