In Algebraic Topology, profinite homotopy types are frequently encountered. This is often because of the use of profinite completion?s of homotopy types in an attempt to get more ‘accessible’ information out of a homotopy type. The theory has applications both in Algebraic Geometry and Algebraic topology.
One origin of the theory can be found in Grothendieck's Galois theory, in which he defined an algebraic fundamental group of a scheme using its finite ‘covering spaces’. These correspond to the finite field extensions in the The classical case of fields, and from that perspective one can ask what the higher profinite homotopy n-types of a scheme should classify.
In the 1960s Artin and Mazur constructed a functor which associates to each locally noetherian scheme its étale homotopy type?, , an object of , the pro-category of the homotopy category of simplicial sets. They observed that this did not correspond to a homotopy category of a model category on a category of pro simplicial sets. (This observation was the start of the search for a suitable pro-homotopy theory.)
Friedlander gave a rigidified version of the Artin-Mazur homotopy type, which he called the étale topological type? of the scheme. This was used by Quillen in the proof of the Adams conjecture, a result purely in Algebraic Topology.
Dennis Sullivan introduced profinite completions into topology in his work: D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math., 100, (1974), 1–79. The Adams conjecture, which is a statement about purely topological phenomena, was then proved by Quillen using these ideas.
At about the same time, Bousfield and Kan, studied profinite completions amongst a wealth of other stuff in A. Bousfield and D. Kan, 1972, Homotopy limits, Completions and localizations, volume 304 of Lecture Notes in Maths , Springer-Verlag.
In the 1990s, Morel and Voevodsky defined a neat framework for the use of topological methods in algebraic geometry. They embedded the category of smooth schemes of ‘finite type over a field into a larger category of ’-spaces’, which carries the structure of a closed model category. The study of these -spaces is linked to étale homotopy theory, see Schmidt, On the étale homotopy type of Morel-Voevodsky spaces, and Dan Isaksen, Etale realization on the -homotopy theory of schemes. Adv. in Math. 184, 37–63 (2004).
An elegant and well structured theory has been published by Gereon Quick, in order to give a good profinite completion construction; (see G. Quick, Profinite homotopy theory, Documenta Mathematica, 13, (2008), 585–612, (Archiv 0803.4082) and Some remarks on profinite completion of spaces , Arxiv preprint arXiv:1104.4659.