ScalingStacks

Theorem 8.3. Let ${\operatorname{\mathbf{M}}}$ be simplicial closed model category, and let ${\operatorname{\mathbf{W}}}\subset{\operatorname{\mathbf{M}}}$ denote the subcategory of weak equivalences. Then $V=N^{f}({\operatorname{\mathbf{M}}},{\operatorname{\mathbf{W}}})$ is a complete Segal space. Furthermore, there is an equivalence of categories $\ho V\approx\ho{\operatorname{\mathbf{M}}}$ and there are weak equivalences of spaces $\map_{V}(X,Y)\approx\map_{{\operatorname{\mathbf{M}}}}(X,Y)$.