Remark 9.2.4.2. Definition 9.2.4.1 has a counterpart in classical category theory. If $\operatorname{\mathcal{D}}$ is an ordinary category, we say that a functor $\mathscr {F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}$ is flat if the category of elements $\int _{\operatorname{\mathcal{D}}} \mathscr {F}$ is cofiltered (Definition 5.2.6.1). It follows from Corollary 9.1.2.8 that this is condition is satisfied if and only if the nerve
is a flat functor of $\infty $-categories, in the sense of Definition 9.2.4.1. We will see below that, up to isomorphism, every flat functor $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}$ can be obtained in this way (Remark 9.2.4.18). Consequently, Definition 9.2.4.1 can be regarded as a generalization of its classical counterpart.