Some remarks on profinite completion of spaces

TL;DR

This paper develops a comprehensive theory of profinite completion for spaces, introducing new functors and methods that extend existing concepts to non-connected and equivariant settings, with applications in Galois theory and Teichmüller theory.

Contribution

It constructs a rigidification of Artin-Mazur and Sullivan's profinite completion functors, extending them to non-connected and equivariant contexts, and surveys key results in the field.

Findings

Introduced an equivariant profinite completion functor.

Extended completion functors to non-connected spaces.

Provided a survey of known results on profinite completion of spaces.

Abstract

We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of Artin-Mazur and Sullivan which extends also to non-connected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications where, for example, Galois groups are involved, or for profinite Teichmueller theory where equivariant completions are applied. Along the way we collect and survey the most important known results about profinite completion of spaces.