ScalingStacks

Proof. The proof of this theorem follows from a refinement of the proof of Theorem 2.2. The model structure therein arises as the Bousfield localization of a cofibrantly-generated model structure on $\Cat_{\mathcal{S}}$ in which the weak equivalences are the levelwise equivalences [74, §4], i.e., the spectral functors $F:{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ such that for all objects $x,y\in{\mathcal{A}}$, the morphism ${\mathcal{A}}(x,y)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}(Fx,Fy)$ is a levelwise equivalence, and the induced simplicial functor $\Omega^{\infty}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}({\mathcal{B}})$ is a DK-equivalence.

First, we observe that the category $\Cat_{\mathcal{S}}$ is locally presentable; a set of small generators is given by applying the functor $U$ (see [74, A.1]) to a set of small generators for the category of symmetric spectra. Since the leverwise model structure on $\Cat_{\mathcal{S}}$ is cofibrantly generated, it follows that it is combinatorial. Next, the arguments of [74] produce a generating set of DK-equivalences at which to localize $\Cat_{\mathcal{S}}$. The main theorem about the existence of left Bousfield localization for combinatorial model categories (e.g., see the treatment in [2]) now implies that we can localize and obtain a combinatorial model structure on $\Cat_{\mathcal{S}}$.

The machinery of Dugger’s approach to universal homotopy theories [24] now permits us to replace $\Cat_{\mathcal{S}}$ with a Quillen equivalent simplicial model category (the simplicial objects over $\Cat_{\mathcal{S}}$) which is combinatorial and left proper. By applying the techniques of [24, 66], we can promote this adjunction to a simplicial Quillen adjunction. Specifically, the simplicial prolongation of the adjunction forms a Quillen pair on the categories of simplicial objects [66, 6.1]. ∎