## Profinite homotopy theory

### Summary

Summary: 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 étale homotopy theory of schemes in which higher profinite étale homotopy groups fit well with the étale fundamental group which is always profinite. We show that the profinite étale topological realization functor is a good object in several respects.

### Mathematics Subject Classification

55P60, 55Q70, 14F35

### Keywords/Phrases

profinite completion, profinite homotopy groups, étale homotopy type of schemes