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Let $G$ be a profinite group (defined as a Hausdorff, compact, totally disconnected topological group). Suppose you know that as a profinite set, it's an inverse limit of finite sets. Is there an abstract category-theoretic way to deduce $G$ is an inverse limit of finite groups, or must the proof have more content?

I know how to prove it using more content, but I've been advised (if I understood correctly) that there's a simple abstract reason. I can't work out what it is.

Eric Wofsey
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gothicvague
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Jun 18 '22 at 15:22
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    There is a lot of information about when the two senses of "profinite algebra" agree in Johnstone's book Stone Spaces, Chapter 8 if I remember right. – Alex Kruckman Jun 18 '22 at 18:43
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    @AlexKruckman: There is indeed, but it is in Chapter 6, specifically section VI.2. In particular VI.2.8 and VI.2.9 give some (fairly complicated) syntactic criteria for the two senses of "profinite algebra" to agree. – Eric Wofsey Jun 18 '22 at 19:03
  • @EricWofsey Thanks for correcting my reference! I didn't have my copy of the book close to hand. – Alex Kruckman Jun 18 '22 at 19:06

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As evidence that you cannot expect an "abstract nonsense" proof of this, the corresponding statement for Jónsson-Tarski algebras is false. A Jónsson-Tarski algebra is a set $X$ together with a binary operation $w:X\times X\to X$ and a pair of unary operations $p,q:X\to X$ such that $w$ and $(p,q):X\to X\times X$ are inverse to each other. So, a topological Jónsson-Tarski algebra can be described as a topological space $X$ together with a homeomorphism $X\times X\to X$. In particular, for instance, $\{0,1\}^\mathbb{N}$ is a profinite set which admits a topological Jónsson-Tarski algebra structure. However, $\{0,1\}^\mathbb{N}$ cannot be an inverse limit of finite Jónsson-Tarski algebras, since a finite Jónsson-Tarski algebra has at most one element (if $X$ is a finite set with more than one element then $|X\times X|\neq |X|$!).

So, any proof of this statement for groups is going to need to use something special about groups beyond just that they are a type of algebraic structure.

(As Alex Kruckman commented on the question, you can find more examples and discussion of this question in general in Peter Johnstone's book Stone spaces, specifically in section VI.2. In particular, VI.2.8-9 explores some sufficient conditions on a variety of algebras under which you can generalize the usual argument for profinite groups to show that inverse limits of finite algebras coincide with topological algebras whose underlying space is a profinite set.)

Eric Wofsey
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