ScalingStacks

Proof. The category $\ho W$ is a groupoid if and only if $\hoequiv(x,y)=\map(x,y)$ for all $x,y\in{\operatorname{ob}}W$, if and only if $W_{\hoequiv}=W_{1}$, if and only if $s_{0}\colon W_{0}\rightarrow W_{1}$ is a weak equivalence (since $W$ is complete). A simplicial space $W$ is weakly equivalent to a constant simplicial space if and only if $s_{0}\colon W_{0}\rightarrow W_{1}$ is a weak equivalence. ∎