Lemma 7.6 (0NN8) · ScalingStacks

0NN8

Lemma 7.6. Let ${\mathcal{A}}$ be a combinatorial simplicial model category and ${\mathcal{C}}\subset{\mathcal{A}}$ a full subcategory. Then for each $n$ the induced map

\mathrm{N}(({\mathcal{C}}^{\Ar[n]})^{\cf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathrm{N}(\Ar[n]),\mathrm{N}({\mathcal{C}}^{\cf}))

is a categorical equivalence of simplicial sets.

0NN9

Proof. By [52, A.3.4.15], we can choose a small subcategory ${\mathcal{V}}\subset{\mathcal{A}}$ which contains ${\mathcal{C}}$ and such that ${\mathcal{V}}$ is an $(\Ar[n])$ -chunk for each $n$ and moreover $\mathrm{N}(({\mathcal{C}})^{\cf})$ is equivalent to $\mathrm{N}(({\mathcal{V}})^{\cf})$ . Then as discussed above, [52, 4.2.4.4] implies that for each $n$ the natural map

\mathrm{N}((({\mathcal{V}})^{\Ar[n]})^{\cf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathrm{N}(\Ar[n]),\mathrm{N}(({\mathcal{V}})^{\cf}))

is a categorical equivalence of simplicial sets. ∎