Singular functors and realization functors

AbstractIn [6] Quillen showed that the singular functor and the realization functor have certain properties which imply the equivalence of the weak homotopy theory of topological spaces with the homotopy theory of simplicial sets. The aim of this note is to generalize this result and to show that one can, in essentially the same manner, establish the equivalence of other homotopy theories (e.g., the equivariant homotopy theories) with homotopy theories of simplicial diagrams of simplicial sets. Applications to equivariant homotopy will be given in [3] and [4]