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The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not exactly a wealth of references about profinite sets and profinite completions of sets.

edited Jun 6 at 15:47

PaPiro
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asked Mar 2 2010 at 20:59

jackie boy
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What do you mean by the profinite completion of a set? – Qiaochu Yuan Mar 2 2010 at 21:33

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I guess one could define the profinite completion of a set as the projective limit of all finite sets endowed with a (surjective) map from the given set. – Leonid Positselski Mar 2 2010 at 21:55

Let us first say what a profinite set is. This is a compact Haussdorf totally disconnected topological space. We may form the category of profinite spaces where the morphisms are continuous maps between them. Their is a forgetfull functor from profinite sets to sets that forgets the topology. Profinite completion is the left adjoint to this functor. – jackie boy Mar 2 2010 at 22:01

The projective limit definition would amount to the same thing. – jackie boy Mar 2 2010 at 22:02

Leonid Positselski · Accepted Answer · 2010-03-02 22:38:32Z UTC

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Profinite sets are just another name for compact totally disconnected topological spaces. I think this is (essentially) explained somewhere in Bourbaki's books on general topology.

edited Mar 2 2010 at 22:38

answered Mar 2 2010 at 21:54

Leonid Positselski
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What's an \ell-space? – Leonid Positselski Mar 2 2010 at 22:09

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An $\ell$-space is the terminology used by Bernstein for a locally compact, totally disconnected, Hausdorff (maybe this last adjective is redundant) topological space. It seems that fpqc has been reading up on $p$-adic groups lately - I'll have to talk to DeBacker about this! ;) – Marty Mar 2 2010 at 22:39

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@fpqc: this is not Twitter. If you have something to say, please use whole sentences, and explain how your comment is relevant (are you pointing out an error in Leonid's answer? giving an alternate description? etc.) – Reid Barton Mar 2 2010 at 22:44

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Leonid, you either forgot "Hausdorff" or you were being a bit merciless with your terminology :-) Also: another name for profinite spaces is Stone spaces. – Tom Leinster Mar 3 2010 at 0:58

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@Leonid: Only algebraic geometers make that distinction. Do you say quasilocal for "has a unique maximal ideal" and local for "quasilocal + noetherian" like a commutative algebraist? I don't understand why AG people like to make it an issue. We realize that all of the standard texts use "quasicompact", but I doubt that using "quasicompact" instead of "compact" is ever going to catch on outside of AG. Also, it's silly because Hausdorff spaces pretty much never come up in AG. – Harry Gindi Mar 3 2010 at 14:13

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Marty · Answer 2 · 2010-03-02 22:32:44Z UTC

I agree with some of the comments that "profinite set" is not a standard term. But you can certainly look at the category of pro-(finite sets). In other words, begin with the category $Set_f$ of finite sets and functions. Then one can form a category $Pro(Set_f)$ as the projective completion of the category $Set_f$; it is the full subcategory of the category of functors from $Set_f$ to $Set$, consisting of objects isomorphic to projective limits of systems of finite sets. In other words, the objects of $Pro(Set_f)$ are (not necessarily representable) functors from $Set_f$ to $Set$, which are inductive (viewing finite sets via Yoneda as functors from $Set_f$ to $Set$ switches arrow directions) limits of representable functors from $Set_f$ to $Set$. I'm sure one should be careful about some smallness/universe issues to make this precise.

The category $Pro(Set_f)$ is equivalent to the category of compact totally disconnected topological spaces. This elaborates on Leonid's answer.

The reference for this somewhat highbrow answer is a paper of Gaitsgory and Kazhdan, in GAFA, titled "Representations of algebraic groups over a 2-dimensional local field".

Benjamin Steinberg · Answer 3 · 2011-06-27 01:33:54Z UTC

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Johnstone's book Stone Spaces is a suitable reference for this and much more. The book by Ribes and Zalesskii Profinite Groups also has good information. Algebre et Théories Galoisienne by Doady and Doady also gives a nice accounting.

answered Jun 27 2011 at 1:33

Benjamin Steinberg
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David Corwin · Answer 4 · 2010-07-04 12:14:59Z UTC

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See http://math.harvard.edu/~waffle/boolean.pdf for notes on Boolean algebras and the fact that they are essentially (anti-)equivalent to totally disconnected compact Hausdorff spaces.

answered Jul 4 2010 at 12:14

David Corwin
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By the way, these notes were written by an active MO user. – David Corwin Jul 4 2010 at 12:15

What is a reference for profinite sets?

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