# Décalage and Kan's simplicial loop group functor

## Danny Stevenson

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 simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy 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 $\overline{W}$. We will give a new, short, proof of Kan's result that the unit map for the adjunction $G\dashv \overline{W}$ is a weak homotopy equivalence for reduced simplicial sets.

Keywords: simplicial loop group, d\'{e}calage, Artin-Mazur total simplicial set

2010 MSC: 18G30, 55U10

Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787.

Published 2012-12-13.

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