Proposition 15.10 (0MLG) · ScalingStacks

[0MLG]

Proposition 15.10. Let $\mathcal{A}$ and $\mathcal{B}$ be two model categories of $(\infty,n)$ -categories and let $L:\mathcal{A}\leftrightarrows\mathcal{B}:R$ be a Quillen adjunction between them. Then $(L,R)$ is a Quillen equivalence if and only if the left derived functor $\mathrm{N}^{\mathrm{H}}L:\mathrm{N}^{\mathrm{H}}\mathcal{A}\to\mathrm{N}^{\mathrm{H}}\mathcal{B}$ preserves the cells up to weak equivalence.

[0MLH]

Proof. A Quillen equivalence induces an equivalence $\mathrm{N}^{\mathrm{H}}L:\mathrm{N}^{\mathrm{H}}\mathcal{A}\to\mathrm{N}^{\mathrm{H}}\mathcal{B}$ of $\infty$ -categories. By Lemma 10.2 and Lemma 4.8 any such equivalence necessarily preserves the cells up to equivalence. Conversely, as the left-derived functor $\mathrm{N}^{\mathrm{H}}L:\mathrm{N}^{\mathrm{H}}\mathcal{A}\to\mathrm{N}^{\mathrm{H}}\mathcal{B}$ preserves (homotopy) colimits and $\mathrm{N}^{\mathrm{H}}\mathcal{A}$ and $\mathrm{N}^{\mathrm{H}}\mathcal{B}$ are generated under (homotopy) colimits by the cells (Axiom C.2), it follows that $\mathrm{N}^{\mathrm{H}}L$ induces an equivalence of $\infty$ -categories. In particular it induces an equivalence of homotopy categories, and hence $(L,R)$ is a Quillen equivalence. ∎