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S. carmeli
S. carmeli
07:15
@TimCampion just adding on @PiotrPstrągowski, answer, the analogy is more like Kn modules to Fp modules. Kn local spectra is like the completion around the stratum of height n and you want things which are actually "on it" so you need to take modules I believe to get a somewhat reasonable analogy.
 
2 hours later…
Bruno Stonek
Bruno Stonek
09:27
does somebody have an $\infty$-categorical reference for the fact that the projection of an undercategory $C_{c/}\to C$ creates connected colimits?
Alexander Campbell
Alexander Campbell
@BrunoStonek I think undercategory projections only create contractible colimits in general, for which see HTT.4.4.2.9.
Bruno Stonek
Bruno Stonek
09:44
oh, hmm. I wasn't expecting that. thanks for the reference!
 
5 hours later…
Dylan Wilson
Dylan Wilson
14:58
@BrunoStonek in general to compute colimits in $C_{c/}$ for $p:K\to C$ you can take the pushout of $c \leftarrow \mathrm{colim}_K c \rightarrow \mathrm{colim}_K p$; when $K$ is contractible the first map is an equivalence, but not in general. From this formula it's also easy to see that you only need $n\pm \epsilon$ connectivity in an $n$-category
 
2 hours later…
Tim Campion
Tim Campion
16:57
@PiotrPstrągowski @S.carmeli Thanks, I was being really confused. Of course the analogy should have $K(n)$-modules being compared to $H\mathbb F_p$-modules. I guess something I still find funny is that although people talk about $K(n)$-local homotopy theory all the time with $n$ finite, I rarely hear people talk about $H\mathbb F_p$-local homotopy theory. I wonder why that is?
Denis Nardin
Denis Nardin
@TimCampion Derived completion doesn't work very well in infinite codimension (i.e. when the open complement of the closed subscheme you're completing at is not q-compact). That said $H\mathbb{F}_p$-completion occasionally shows up (e.g. in the convergence of the Adams SS)

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