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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.
@article{TAC_2012_26_a27,
author = {Danny Stevenson},
title = {D\'ecalage and {Kan's} simplicial loop group functor},
journal = {Theory and applications of categories},
pages = {768--787},
publisher = {mathdoc},
volume = {26},
year = {2012},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2012_26_a27/}
}
Danny Stevenson. Décalage and Kan's simplicial loop group functor. Theory and applications of categories, Tome 26 (2012), pp. 768-787. http://geodesic.mathdoc.fr/item/TAC_2012_26_a27/