ScalingStacks

Corollary 7.7. Let ${\mathcal{A}}$ be a simplicial model category and ${\mathcal{C}}\subset{\mathcal{A}}$ a small full subcategory which has all finite homotopy colimits. Then for each $n$ there is a weak equivalence of simplicial sets

$$|w_{\bullet}S^{\prime}_{n}{\mathcal{C}}|\simeq|(S_{n}^{\infty}\mathrm{N}(({\mathcal{C}})^{\cf}))_{\mathrm{iso}}|.$$

and for each $(n_{1},\dotsc,n_{q})$ there is a weak equivalence of simplicial sets

$$|w_{\bullet}S^{\prime(q)}_{n_{1},\dotsc,n_{q}}{\mathcal{C}}|\simeq|((S^{\infty})^{(q)}_{n_{1},\dotsc,n_{q}}\mathrm{N}(({\mathcal{C}})^{\cf}))_{\mathrm{iso}}|.$$