Profinite homotopy theory

Abstract

We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the \'etale homotopy theory of schemes in which higher profinite \'etale homotopy groups fit well with the \'etale fundamental group which is always profinite. We show that the profinite \'etale topological realization functor is a good object in several respects.