ScalingStacks

Remark 14.1. This completion is a generalization of the classifying space construction. In fact, suppose $W$ is a Segal space such that $\ho W$ is a groupoid; equivalently, that $W_{1}=W_{\hoequiv}$. Then the arguments below show that $\widehat{W}$ is weakly equivalent to a constant simplicial space, which in each degree is the realization $\diag W$. For instance, if $W$ is a “$\Delta$-space” (i.e., $W_{0}=*$) and thus a model for a loop space with underlying space equivalent to $W_{1}$, then $\widehat{W}$ is equivalent to the constant object which is $BW_{1}$, the classifying space of the “loop space” $W_{1}$, in each degree.