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What is it you are looking at? Do you want to replace in the traditional definition of completion 2-groups/crossed modules for the traditonal abelian groups appearing there?
You might want to talk to Tim Porter about this. He’s written a book about profinite crossed modules. He has a page on the nLab, or find my email at David Roberts and I can send you his details.
You can find the pro-C completion discussed by following Profinite Algebraic Homotopy in my home page:
http://ncatlab.org/timporter/show/HomePage
There is a downloadable document corresponding to an old version of the book that David is referring to above, and it has a discussion of the pro-finite completion, that may be useful.
The initial work was done by Korkes in his thesis and a joint paper in Proc. Edin Math. Soc. 1990.
(BTW The use of ’adic’ as an adjective is awkward. This is really a construction (p-adic, M-adic, polyadic, operadic) whic constructs a new adjective from some other word. Its use in a stand alone position is very strange.)
Since it seems you all know which definition exactly elif is looking for (I can guess what it is, but you all seem to be sure): It shouldn’t be hard to just state the definition here.
@Urs. I am not at all sure what elif is looking for. The point I was making was more a linguistic one. I had noted in several papers by US authors a tendency to talk about adic completions and and have wondered what they were since there is a whole family of completions available.
My ex-student did look at the profinite completion for crossed modules and that extends to a pro-C one. I have wondered if this is the ’right’ completion for other purposes and if I can I fill put something here or, better, on the n-lab about it. The notion of profinite completion of homotopy types is complicated and that is much harder to describe in a deffinitive way….. but in any case, I do not know what an ‘adic’ completion (of anything) is!
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