A model structure via orbit spaces for equivariant homotopy

Abstract

Let G be discrete group and \(\mathcal F\) be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H-orbits for all H in \(\mathcal F\). This gives a model categorical criterion for maps that induce weak equivalences on H-orbits to be weak equivalences in the \(\mathcal F\)-model structure.