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 group*oids* 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?

closedsubgroups 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 6:23more datathan knowing a pro-object in the category of finite groupoids. And if you are a fan of enriched category theory, you might actuallylikestudying groupoids internal to Stone spaces. – Theo Johnson-Freyd Dec 15 '10 at 7:53