Timeline for Why study the p-completions of a space?

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Dec 18, 2019 at 16:48	comment	added	Yonatan Harpaz		One way to define $p$-completion of $X$ is to take the simplicial group $G_{\bullet}$ which models the loop space $\Omega X$ and then $p$-complete it levelwise, so that at level $n$ the group $G_n$ is replaced by the limit of its finite $p$-quotients. One can then show that this is the same as taking the limit over all coskeletal levelwise $p$-finite simplicial groups $H$ equipped with a map $G \to H$. The classifying space of each $H$ is then a $p$-finite space, and we get that the completion of $X$ (and hence $X$ itself if it is $p$-complete) is a limit of $p$-finite spaces.
Dec 18, 2019 at 16:27	comment	added	Saal Hardali		I might have misinterpreted but is there a claim here that every $p$-complete space is the limit of some $p$-profinite space? I thought this might fail for some wild enough non-nilpotent spaces, I didn't know mandell theorem was that general...
Apr 22, 2016 at 19:35	history	edited	Yonatan Harpaz	CC BY-SA 3.0	added 2 characters in body
Apr 22, 2016 at 6:19	comment	added	Ilan Barnea		Great answer! Here are some typos that I was not able to edit: near the end of the first paragraph it should be "$G: Pro(D) \longrightarrow Pro(C)$, when restricted to $D$". Also in the second paragraph: $G$ is localization w.r.t. $Z/p$-cohomology.
Sep 16, 2015 at 11:35	vote	accept	Tyrone
Sep 13, 2015 at 19:38	history	answered	Yonatan Harpaz	CC BY-SA 3.0