ScalingStacks

Remark 8.8. We can interpret (8.7) as saying that for any two fibrant-and-cofibrant objects $X,Y\in{\operatorname{\mathbf{M}}}$, the space of paths from $X$ to $Y$ in $\class({\operatorname{\mathbf{M}}})$ is naturally weakly equivalent to the space $\hoequiv_{{\operatorname{\mathbf{M}}}}(X,Y)\subset\map_{{\operatorname{\mathbf{M}}}}(X,Y)$ of homotopy equivalences from $X$ to $Y$. (The notation $\class({\operatorname{\mathbf{M}}})$ was defined in (1.2).) Compare with (6.4, 4).