  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.
