Kerodon

Remark 4.5.6.2. Definition 4.5.6.1 is a special case of a general convention. If $P$ is a property of morphisms of simplicial sets and $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a natural transformation between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, then we say that $\alpha $ has the property $P$ levelwise if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism of simplicial sets $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ has the property $P$. For example, we say that $\alpha $ is a levelwise weak homotopy equivalence if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a weak homotopy equivalence of simplicial sets.