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Created progroup, with remarks about the equivalence between surjective progroups and prodiscrete localic groups.
Why do we have separate pages profinite space and Stone space which do nothing but point to each other? Is there any reason not to merge them?
Oh, and along the way I created posite.
The possible justification for having two pages is that the contexts in which they live are not obviously linked. (I think that there is more of a link than the obvious one.) The contexts are profinite spaces which rubs shoulders with lots of geometry and topology, whilst Stone spaces is linked more to Boolean algebras logic, etc.
Question at posite regarding double negation topology and forcing.
@Tim: hmm. I’m not entirely convinced, especially given that both are currently stubs. Could we merge them for now, and if later on it turns out that the material is bifurcating, split them up again?
@David: I replied: yes!
@Mike Just saw your nice comment on the Café and am trying to understand it. The reason I am interested (apart from the intrinsic interest of course) is that years ago I had a Postgrad, Fahmi Korkes, who looked at profinite completions of crossed modules and the general theory of profinite crossed modules. (We published a couple of short notes derived from his thesis. He was unable to write up more due to various wars etc. No need for explanation here.) I have made available a much larger draft which grew out of his work (see http://ncatlab.org/timporter/show/profinite+algebraic+homotopy. That version is not complete as it may be going to be published. )
There is an interesting point that a referee for that ’book’ made relating to Dan Isaksen’s paper (Calculating limits and colimits in pro-categories, Fund. Math. 175 (2002) 175-194.) The oint is that pro- simplicial finite sets and simplicial profinite sets seem to be non-equivalent categories. (I think the point is a sort of ’phantom’ behaviour towards infinity. I am not sure what I mean by that :-(! I think I checked that the n-truncated simplicial objects are equivalent but that the equivalence may take more and more ’reindexing’ as you go up the dimensions. I am so far unable to see if this makes a big difference to the representability of the profinite homotopy types. I convinced myself that for the ’material’ I was looking at there was unlikely any problem.
Another point perhaps worth making is that there is work by Quick (Profinite homotopy theory, Doc. Math. 13 (2003) 585-612) which gives some results that may be of interest, especially on profinite completions of ’spaces’. (I have yet to read his papers … there are others… in detail, having just skimmed them on-line. This area is big at the moment as it relates to motives etc, which may be relevant to your posting as well! .. I do not understand what the motive for motives is :-) (Yeah it was old and weak as a joke! )
Hi Mike,
I edited your progroup slightly. Made the definition of surjective progroup a separate definition and made the first line of your proof of the theorem point pack to the previous definition.
Thanks.
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