Timeline for When $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$?

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Jul 27, 2020 at 20:02	comment	added	Ben Wieland		When in doubt interpret $\Sigma^\infty(X_p)$ as $(\Sigma^\infty X)_p$. Or, rather, interpret it as $(\Sigma^\infty X_p)_p$, which is equivalent to $(\Sigma^\infty X)_p$. So this is a functor that you can apply to $p$-complete spaces. (How generally are they equivalent? If you're using $\mathbb Z/p$-localization, it's easy to see that they're always equivalent. If you're using $p$-completion, I'm not sure if it's always equivalent, but usually you're using it because in that case it's equivalent to $\mathbb Z/p$-localization.)
Jul 16, 2020 at 7:24	comment	added	Denis Nardin		@VictorTC Unfortunately those kinds of notations are wildly inconsistent from paper to paper (especially when you go to somewhat old ones). There's not much to do but be extra careful.
Jul 15, 2020 at 22:20	comment	added	Victor TC		@skd Thanks, that article proves that $\Sigma^{\infty} (BS^{1})^{\wedge}_p$ is not $p$-complete, what made me think that $BG^{\wedge p}$ in the stable homotopy context means $(\Sigma^{\infty} BG)^{\wedge p}$, although it seems counterintuitive to me.
Jul 15, 2020 at 18:24	comment	added	skd		See arxiv.org/abs/1712.07633
Jul 15, 2020 at 17:45	history	edited	Victor TC	CC BY-SA 4.0	added 125 characters in body; edited title
Jul 15, 2020 at 16:37	history	asked	Victor TC	CC BY-SA 4.0