A version of an E 2 closed model category structure

Abstract

Let X • be a pointed simplicial set. Its constant bisimplicial extension cX •• can be thought of as a resolution of X • by the zero spheres in the following sense: 1) by “flipping the axes”, cX • • can be seen as a simplicial object having in each degree a wedge of zero spheres; 2) the diagonal Δ(cX • •) is isomorphic to X •. Replacing S 0 with seine pointed simplicial set A • one can ask what is the best possible approximation of X • by a diagonal of a simplicial set having in all of its degrees wedges of A • and its suspensions, and what is the relation of this approximation to CW A X •, the A •-colocalization of X •. In this article we construct a closed model category structure in which such a resolution can be obtained by factoring the unique map from the basepoint into the given bisimplicial object as a cofibration followed by an acyclic fibration. These resolutions can be used to construct spectral sequences whose E 2-term depends on A •-homotopy of X • and ordinary homotopy of A • and which converge to the ordinary homotopy of the best possible approximation of X •by homotopy colimits of diagrams whose objects are coproducts of A • and its suspensions.