The functors \(\overline {W}\) and \(\text{Diag}\circ \text{Nerve}\) are simplicially homotopy equivalent. (Q847575)

Let \(G\) be a simplicial group. There are two well-known classifying simplicial set constructions: (1) Kan's classifying simplicial set \(\overline {W}G\) and (2) dimensionwise application of the nerve functor for groups yields a bisimplicial set \(NG\), to which one can apply the diagonal functor to obtain a simplicial set \(\operatorname{Diag} NG\). It is well-known that \(\overline {W}G\) is weakly homotopy equivalent to \(\operatorname{Diag} NG\). In this article, the author proves that \(\overline {W}G\) is a strong simplicial deformation retract of \(\operatorname{Diag} NG\). This gives a stronger relationship between \(\overline {W}G\) and \(\operatorname{Diag}NG\).