The functors Wbar and Diag o Nerve are simplicially homotopy equivalent

TL;DR

This paper proves that the classifying simplicial sets Wbar G and Diag N G for a simplicial group G are not only weakly homotopy equivalent but are actually simplicially homotopy equivalent, with Wbar G being a strong deformation retract of Diag N G.

Contribution

It establishes a strong simplicial deformation retraction between Wbar G and Diag N G, strengthening the known weak homotopy equivalence to a simplicial homotopy equivalence.

Findings

Wbar G is a strong simplicial deformation retract of Diag N G.

Wbar G and Diag N G are simplicially homotopy equivalent.

The result clarifies the relationship between two classifying simplicial set constructions.

Abstract

Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set Wbar G and Diag N G, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We will show that Wbar G is a strong simplicial deformation retract of Diag N G. In particular, Wbar G and Diag N G are simplicially homotopy equivalent.