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I noted the point made by, I think, Toby about there being stuff on profinite homotopy type in the wrong place (profinite group). I have started up a new entry on profinite homotopy types, but am feeling that it needs some more input of ideas, so help please.
Recent paper by Hess has some new viewpoints on completions, among other ideas, in very general setup. It may be of remote interest here.
That raises the problem of the relationship at a fairly basic geometric level of the completion stuff in algebraic topology, with the more abstract ideas in quasicategories etc. Perhaps some entries on the completion theory and its relationship to localisation theory will be needed sometime.
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