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.
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$\begingroup$ What do you mean by the profinite completion of a set? $\endgroup$– Qiaochu YuanCommented Mar 2, 2010 at 21:33
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2$\begingroup$ 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. $\endgroup$– Leonid PositselskiCommented Mar 2, 2010 at 21:55
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1$\begingroup$ 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. $\endgroup$– jackie boyCommented Mar 2, 2010 at 22:01
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$\begingroup$ The projective limit definition would amount to the same thing. $\endgroup$– jackie boyCommented Mar 2, 2010 at 22:02
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4 Answers
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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.
answered Mar 2, 2010 at 21:54
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14$\begingroup$ @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.) $\endgroup$ Commented Mar 2, 2010 at 22:44
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3$\begingroup$ Leonid, you either forgot "Hausdorff" or you were being a bit merciless with your terminology :-) Also: another name for profinite spaces is Stone spaces. $\endgroup$ Commented Mar 3, 2010 at 0:58
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3$\begingroup$ @Tom: In the terminology I am used to, "compact" presumes Hausdorff (when it doesn't, the term "quasicompact" is being used). $\endgroup$ Commented Mar 3, 2010 at 10:51
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3$\begingroup$ Compact non-Hausdorff spaces are important, though perhaps not in your specific field. As I was saying, "compact presumes Hausdorff" is not conventional enough to use it without note, whence Tom's remark. $\endgroup$ Commented Mar 3, 2010 at 18:51
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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".
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$\begingroup$ More precisely, objects of $Pro(Set_f)$ are covariant functors from $Set_f$ to $Set$ which are filtered inductive (rather than projective) limits of representable functors. $\endgroup$ Commented Mar 2, 2010 at 22:27
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$\begingroup$ Whoops - I forgot about the Yoneda reversal. I think it's correct now. Thanks Leonid! $\endgroup$– MartyCommented Mar 2, 2010 at 22:33
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$\begingroup$ This is a little highbrow for my taste. [Marty is an old friend of mine, so I know he can take being called "highbrow" with a smile.] Suppose that $S$ is a countably infinite set. Its profinite completion is...? $\endgroup$ Commented Mar 3, 2010 at 2:34
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$\begingroup$ Excuse my high brows :) I didn't say anything about profinite completions -- I don't know a definition of "the profinite completion of a set". Given a set $X$, endowed with the discrete topology, perhaps the Stone-Cech compactification $\beta X$ satisfies an appropriate universal property to be called a "profinite completion of $X$". I wouldn't use this terminology though! "Profinite completion" should only be used for groups, I think. $\endgroup$– MartyCommented Mar 3, 2010 at 4:11
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$\begingroup$ What I am calling the profinite completion of a set is almost the same as for a group, except everytime the word group is used, you replace it the with the word set. Now the Stone Cech compactifaication of a (discrete) space misses the words, "totally disconnected". Wikipedia states (so a grain of salt) that the stone cech compactification happens to be totally disconnected. So ths notion of profinite completion seems to be the stone cech compactification. I would like to make a remark on the terminology. Any concrete category with filtered limits has a "profinite completion" functor. $\endgroup$ Commented Mar 3, 2010 at 6:03
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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 Douady and Douady also gives a nice accounting.
answered Jun 27, 2011 at 1:33
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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
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$\begingroup$ By the way, these notes were written by an active MO user. $\endgroup$ Commented Jul 4, 2010 at 12:15