The relationship between the diagonal and the bar constructions on a bisimplicial set

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set W¯X, the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg-Zilber theorem of Dold-Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.