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I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological spaces). This equivalence carries over to profinite vs Stone-topological algebras for a number of algebraic theories, such as groups, monoids, semigroups, and rings. The case of profinite groups is especially well-known.

My question is: why are such equivalences important? Where in mathematics do we gain something by identifying a pro-(finite group) with a Stone topological group? I mean something other than "concreteness" or "familiarity"—certainly it may be easier (for some people) to think about a Stone topological group than about a cofiltered diagram of finite groups, but are there important things that we couldn't prove about cofiltered diagrams of finite groups without knowing that they are equivalent to Stone topological groups?

I am especially interested because this manifestation of Stone duality seems to be "fragile" for generalizations in several directions. For instance, Theo JF commented on this question that Stone-topological groupoids are not equivalent to pro-(finite groupoids). The equivalence is also false if we generalize from finite sets/groups to ones of larger cardinality. It is true that pro-groups with surjective transition maps can be identified with pro-discrete locales, but I don't know anything about whether this is true for pro-sets (cf. question linked above), or any type of algebras other than groups. So in all the cases where the generalization fails, what is lost if we just work with pro-objects and ignore the missing topological aspect?

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I am not sure if that's the sort of answer you are looking for, so I will post it as a comment. The topology often gives a convenient language to express that something happens "on the finite level". E.g. if you consider an infinite Galois extension of fields, its Galois group is pro-finite and closed subgroups correspond bijectively to intermediate fields under the Galois correspondence. Of course, you can express this using co-filtered diagrams, but it would be harder to parse and harder to get a feel for the statement. – Alex B. Dec 15 '10 at 6:23
Not just Galois groups of infinite-degree extensions, but such basic groups such as the additive group $\mathbf{Z}_p$ or the multiplicative group $\mathbf{Z}_p^\times$ are profinite groups. – Chandan Singh Dalawat Dec 15 '10 at 6:32
@Chandan Yes, I was trying to give an example of where the topology actually matters, i.e. where it is substantially easier and more intuitive to formulate an important and natural statement in the topological language than in the language of co-filtrations. – Alex B. Dec 15 '10 at 6:46
Incidentally, there are at least two "Theo"s on this site interested in category theory and related topics. (The other one recently answered a question of mine.) – Theo Johnson-Freyd Dec 15 '10 at 7:38
Here are two partial answers. If you think either is good enough for an actual answer, I can repost them, but I don't think they're worth it. (1) As Alex Bartel suggested, it's convenient to use "topological" intuition when handling pro-objects. (2) At least in the profinitegroupoid lack-of-associativity, knowing a groupoid internal to Stone spaces is more data than knowing a pro-object in the category of finite groupoids. And if you are a fan of enriched category theory, you might actually like studying groupoids internal to Stone spaces. – Theo Johnson-Freyd Dec 15 '10 at 7:53

The answer is "for lots of reasons". Let me explain a couple:

1. As several people have already noted in answers and comments, sometimes profinite groups arise naturally. For example, if $k$ is a field and $k^s$ is its separable closure, it is natural to consider the group of automorphisms of $k^s$ over $k$. One then equips this group with a topology (the weak topology, when $k^s$ is equipped with its discrete topology --- i.e. two elmenents are close it their action coincides on a large finite set of elements of $k^s$), and then discovers that this makes the automorphism group into a profinite group.

Now one can bring the pro-structure to the fore by regarding $k^s$ not just as a field extension of $k$, but as an ind-finite extension of $k$, by writing it as the inductive limit of its finite subextensions. But, while this is technically useful in some contexts (for example, in the proof the automorphism group is profinite), it is not always convenient --- there are often advantages to having $k^s$ available as a naked field, without having to bother with its ind-structure.

2. The concept of topology is incredibly, amazingly flexible, much more so than the concept of pro-object. There are lots of illustrations of this, but one very convincing one is the theory of the adeles. Here one takes the topological product of a profinite ring with copies of $\mathbb R$ and $\mathbb C$. One obtains a locally compact ring, equips it with a Haar measure, and proceeds to do harmonic analysis. Trying to carry all this out in the language of pro-systems (say, of pro-Lie groups) would be incredibly convoluted. Indeed, in the early days of class field theory, before the introduction of the adelic view-point, this is essentially what people did: they worked explicitly with the pro-systems underlying the adeles (without using that language, of course). The introduction of ideles and adeles swept away the inherent (conceptual and notational) complexities of that view-point, and so was (and is) rightly regarded as a major advance.

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As a variant on #2, Hilbert et al. expressed the local-to-global principle for quadratic forms over number fields in terms of big systems of congruences, and recognizing these as equalities in a ($p$-adic) field gave access to powerful tools such as the Galois theory of those fields. Even in $\ell$-adic etale cohomology, where the "sheaves" are pro-systems (in a suitable formalism), passing to profinite inverse limits on stalks is very effective in proofs. And try to express faithful flatness of completion of local noetherian rings in terms of pro-systems. Limit objects have good structure. – BCnrd Dec 15 '10 at 17:47
Why is "one equips this group with a topology" a more natural thing to do than "one considers an inverse system of which this group is a limit"? I admit it may be more comfortable, familiar, or convenient, but it seems like in either case one is choosing an additional structure to put on the object which happens to be convenient for a particular purpose. And in the profinite case, of course, the two types of "structure" are completely equivalent! – Mike Shulman Dec 15 '10 at 19:29
Dear Mike: When limit objects exist they can have properties that are hopeless to express in terms of ind or pro-objects, as indicated in my first comment above. It's used deeply in EGA IV for injlim's: apply the theory to a limit & descend! And whereas "discrete" cohomology of a profinite Galois group is equivalent to injlim of cohomologies at "finite level", it's crucial (e.g., in $p$-adic Hodge theory) to consider "continuous" cohomologies. E.g., for $p$-adic field $K$ and completed alg. closure $C_K$, Tate's vanishing results on $H^i(G_K,C_K(j))$ are hopeless to express at "finite level". – BCnrd Dec 15 '10 at 21:50
Dear Mike: It may clarify the issue of convincing important examples of profinite topologies if you describe your experience with seeing profinite topologies used to prove things that are interesting (to you) and not specifically about profinite topologies. Number theory and arithmetic geometry provide many examples (as above), going way beyond merely being convenient language. But do you seek something in a different area of math? Avoiding the concept of profinite topologies feels like doing ODE's in the language of Dedekind cuts: it is necessary in order to express and develop deep ideas. – BCnrd Dec 15 '10 at 23:27
Dear David: The topology captures the interesting representations and cohomology of the group. The motivation comes from how the groups are used, much as for Lie groups (i.e., we don't primarily study their representation theory as abstract groups, but rather mix it up with the topological and even differentiable/analytic structure). Ultimately we have to know why we care about the group at all. If we only do superficial things with the topology, it's language. But Tate's $p$-adic cohomology results and faithfully flat descent from completions are not reasonably expressed in pro-language. – BCnrd Dec 16 '10 at 0:57

As opposed to a pro-group, a topological group has the chance of being the automorphism group of some mathematical object.

- The automorphism group of an algebraic closure of $\mathbb Q$.
- the group of automomorphisms of a regular rooted tree.
- the group of automorphisms of a 1-dimensional abelian formal group over $\mathbb F_p$

Those mathematical objects have a lot of interesting countable groups acting on them. Thinking in terms of pro-groups, one might forget to investigate those countable groups.

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In what sense "is" the automorphism group of a mathematical object topological any more than it is pro-? As I said in my comment to Emerton's answer, it seems like a choice of topology is analogous to a choice of inverse system. – Mike Shulman Dec 15 '10 at 19:30
For a prorepresentable functor $F:C \to Set$ (as in Grothendieck's Galois theory) there is a natural progroup acting on $F$ in the functor category. – David Roberts Dec 15 '10 at 23:26

Compact semigroups have nice properties (existence of idempotents, minimal ideals, etc). If you view a profinite semigroup as a compact semigroup you get all this for free. I am not even sure how to think of these things for a pro-object in the category of finite semigroups.

Also one wants to think about free profinite groups and semigroups as missing free objects from the categories of finite groups and semigroups. That is the categories of profinite groups and semigroups have natural forgetful functors to sets which have left adjoints. It is not so natural from the pro-object viewpoint.

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