ScalingStacks

Proof. There is an “inclusion” functor ${\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L^{H}{\mathcal{C}}$. Restricting to the weak equivalences and passing to nerves via $\mathrm{N}$, we obtain a map of simplicial sets

(2.11)

$$\mathrm{N}(w{\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(L^{H}w{\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}((L^{H}w{\mathcal{C}})^{\mathrm{fib}});$$

note that the nerve of $w{\mathcal{C}}$ is the same whether we regard it as a category or as a category (trivially) enriched in simplicial sets [52, 1.1.5.8]. Since $(\mathrm{N}((L^{H}w{\mathcal{C}})^{\mathrm{fib}}))_{\mathrm{iso}}$ is isomorphic to $\mathrm{N}((L^{H}w{\mathcal{C}})^{\mathrm{fib}})$, the inclusion $L^{H}w{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L^{H}{\mathcal{C}}$ induces a natural map $\mathrm{N}((L^{H}w{\mathcal{C}})^{\mathrm{fib}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathrm{N}((L^{H}{\mathcal{C}})^{\mathrm{fib}}))_{\mathrm{iso}}$; under the hypothesis that ${\mathcal{C}}$ satisfies a homotopy calculus of fractions, this map is a weak equivalence [29, 6.4]. Therefore, it suffices to show that the map of equation 2.11 is a weak equivalence. We consider the map on components; for each homotopy equivalence class $[x]$, both sides are equivalent to $B\haut(x)$ and it is straightforward to see that the map induces the equivalence. ∎