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

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