Warning 7.2.8.2. Definition 7.2.8.1 has a counterpart in classical category theory. In [MR1815045], Adámek and Rosický define a sifted category to be a nonempty category $\operatorname{\mathcal{C}}$ which satisfies the following condition:
- $(\ast )$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the nerve of the category $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{Y/}$ is connected.
It follows from Corollary 7.2.8.9 below that if the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is sifted (in the sense of Definition 7.2.8.1), then the category $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$. Beware that the converse is false (see Exercise 7.2.8.11). In other words, Definition 7.2.8.1 is not a generalization of the classical notion of a sifted category (instead, it generalizes the notion of a homotopy sifted category, introduced by Rosický in [MR2349711]).