Proposition 2.10 (0NJI) · ScalingStacks

0NJI

Proposition 2.10. Let ${\mathcal{C}}$ be a small category with a subcategory $w{\mathcal{C}}$ of weak equivalences which satisfies a homotopy calculus of two-sided fractions (in the sense of Dwyer and Kan [29, 6.1]). Then there is a weak equivalence of simplicial sets

\mathrm{N}(w{\mathcal{C}})\simeq(\mathrm{N}((L^{H}{\mathcal{C}})^{\mathrm{fib}}))_{\mathrm{iso}},

where here $L^{H}{\mathcal{C}}$ denotes the hammock version of the simplicial localization [29] and $(L^{H}{\mathcal{C}})^{\mathrm{fib}}$ is a fibrant replacement of $L^{H}{\mathcal{C}}$ as a simplicial category.

0NJJ

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. ∎