Kerodon

Example 7.5.9.9. Let $\operatorname{\mathcal{C}}$ be a small category. Then a diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is projectively cofibrant (in the sense of Definition 7.5.6.1) if and only if the unique natural transformation $\underline{\emptyset } \rightarrow \mathscr {F}$ is a projective cofibration (in the sense of Definition 7.5.9.8). Here $\underline{\emptyset }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the initial object of the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, which carries every object of $\operatorname{\mathcal{C}}$ to the empty simplicial set.