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skd
12:07 AM
naive question: how does chromatic height n relate to the n in (oo,n)-category land? i know that there's supposed to be some evidence coming from qft, and maybe relatedly from factorization homology, but i don't understand this very well.
 
 
10 hours later…
10:23 AM
@skd As far as I can tell this should be related to chromatic redshift (e.g. you want to relate K(K(R)) to the K-theory of the (∞,2)-category of modules over Perf(R), see for inspiration Baas-Dundas-Richter-Rognes on 2-vector bundles and K(ku)). I don't know if there's a precise statement hanging around there though
 
 
1 hour later…
11:24 AM
@skd Thanks. I did not know spherical Witt vectors (I know very little on ring spectra). I wonder what we get if we replace the $\Omega^2S^3$ by its truncation $\tau_{\ge2}\Omega^2S^3$? Do we get $W(k)$, just as the case for $k=\mathbb F_p$?
 
 
2 hours later…
1:30 PM
@DenisNardin I think this is very much related to what I was looking for. Thanks.
 
 
1 hour later…
skd
2:38 PM
@DenisNardin ah yes K-theory too! thanks
@FrankScience you do get HW(k) by the same argument
 
 
3 hours later…
6:04 PM
@skd In a slightly different direction there's some evidence for a slightly different interpretation. If we consider $p$-finite spaces ($\pi$-finite all of whose homotopies are $p$-torsion) then there's an obvious filtration by subcategories of $n$-truncated spaces. And for every $n$ the corresponding category embeds fully faithfully in the opposite of the category of $K(n)$-local $E_n$-algebras. This is in Hopkins-Lurie's Ambidexterity paper
 
 
2 hours later…
8:17 PM
@SaalHardali This is stated as a conjecture in Hopkins-Lurie (conjecture 5.4.14). Is it a theorem now?
 
skd
8:49 PM
@SaalHardali thanks
@Marc i heard from jacob that this is indeed a theorem, and unless i'm misremembering, it's proved by explicitly calculating the spectral sequence in rezk's faculty.math.illinois.edu/~rezk/power-ops-ht-2.pdf. not sure of the details though
in fact, i think it's the case that the spectral sequence in 2.14 of rezk's paper already degenerates at the E_2-page
 

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