Equivariant maps which are self homotopy equivalences

Abstract

The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact that the most elementary homotopy theory of “simplicial sets on which a fixed simplicial group H acts” is equivalent to the homotopy theory of “simplicial sets over the classifying complex W ¯ H \bar WH ", and (ii) to use this (in §1) to prove a classification theorem for simplicial sets with an H-action, which provides classifying complexes for their equivariant maps which are self homotopy equivalences.