nLab profinite space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A profinite set is a pro-object in FinSet. By Stone duality these are equivalent to Stone spaces and thus are often called profinite spaces. So these are compact Hausdorff totally disconnected topological spaces.

These are precisely the spaces which are small cofiltered limits of finite discrete spaces, and moreover (as a consequence of Stone duality) the category of Stone spaces is equivalent to the category pro(FinSet)pro(FinSet) of pro-objects in FinSet and finite sets sit FinSetpro(FinSet)FinSet\hookrightarrow pro(FinSet) as finite discrete spaces. This is especially common when talking about profinite groups and related topics.


An internal group in the category of Stone spaces / profinite spaces and continuous maps is a profinite group.


Just as the term ‘space’ is used by some schools of algebraic topologists as a synonym for simplicial set, so ‘profinite space’ is sometimes used as meaning a ‘simplicial object in the category of compact and totally disconnected topological spaces’, i.e. in the other terminology a ‘simplicial profinite space’. This is further complicated by the question of whether or not pro(finite simplicial sets) and simplicial profinite spaces are the same thing.


Profinite reflection


The primary meaning (as Stone space) is used in sources on profinite groups, for which see the entries Stone space, profinite group.

  • Abolfazl Tarizadeh, On the category of profinite spaces as a reflective subcategory (arXiv:1207.5963)

Discussion of homotopy theory of pro(finite) simplicial sets is in

Last revised on November 21, 2013 at 10:13:24. See the history of this page for a list of all contributions to it.