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.
http://www.tac.mta.ca/tac/volumes/26/28/26-28.dvi
http://www.tac.mta.ca/tac/volumes/26/28/26-28.ps
http://www.tac.mta.ca/tac/volumes/26/28/26-28.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/28/26-28.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/28/26-28.ps