ScalingStacks

Proposition 8.7 (Dwyer-Kan [DK84a, 2.3, 2.4]). Given a simplicial closed model category ${\operatorname{\mathbf{M}}}$, and an object $X\in{\operatorname{\mathbf{M}}}$ which is both fibrant and cofibrant, let $\haut X\subset\map_{{\operatorname{\mathbf{M}}}}(X,X)$ be its simplicial monoid of weak equivalences. Then the classifying complex $\bar{W}\haut X$ is weakly equivalent to $\sclass X$; in fact, $\bar{W}\haut X$ and $\sclass X$ can be connected by a finite string of weak equivalences which is natural with respect to simplicial functors $f\colon{\operatorname{\mathbf{M}}}\rightarrow{\operatorname{\mathbf{N}}}$ between closed model categories which preserve weak equivalences and are such that $fX\in{\operatorname{\mathbf{N}}}$ is both fibrant and cofibrant.