D\'ecalage and Kan's simplicial loop group functor

TL;DR

This paper provides a new elementary proof of the weak equivalence between the diagonal and total simplicial sets of bisimplicial sets, and revisits Kan's simplicial loop group functor with a simplified formula and proof.

Contribution

It introduces an elementary proof of a known weak equivalence and offers a simplified formula and proof for Kan's simplicial loop group functor G.

Findings

The comparison map between diagonal and total simplicial sets is a weak equivalence.

A new, simple formula for Kan's simplicial loop group functor G is provided.

The unit map for the adjunction (G,Wbar) is a weak equivalence for reduced simplicial sets.

Abstract

Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these two simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will revisit Kan's simplicial loop group functor G. We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane's classifying complex functor Wbar. We will give a new, short, proof of Kan's result that the unit map for the adjunction (G,Wbar) is a weak equivalence for reduced simplicial sets.