Profinite homotopy theory

TL;DR

This paper develops a model structure on simplicial profinite sets, enabling a profinite completion functor for spaces, with applications to étale homotopy theory of schemes and better understanding of profinite étale homotopy groups.

Contribution

It introduces a new model structure on simplicial profinite sets and constructs a profinite completion functor compatible with étale homotopy theory.

Findings

Constructed a model structure on simplicial profinite sets.

Established a profinite completion functor for spaces.

Applied to étale homotopy theory of schemes.

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.