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12:42 AM
@user40276 no it's false: that colimit needs to be taken in something like the category of presentable categories in order to compute the right answer, so it doesn't even make sense unless C is presentable.
the colimit in (infty-)categories produces the spanier whitehead category, which has the wrong mapping spaces.
(of course, they are related)
 
1:23 AM
@DylanWilson Thanks for the reply. That's weird somehow. I thought that the unit and counit would induce an invertible homotopy between the squares in the (co)limit (co)cones. Still, I should have know that it shouldn't be the case since the colimit will never have non-connective stuff.
 
 
6 hours later…
7:26 AM
@user40276 I think you mean that it cannot contain stuff that's not bounded below :). The SW category is stable, so contains all desuspensions of connective stuff
 
 
2 hours later…
9:01 AM
Is there a framed bordism interpretation of the adams filtration on the homotppy groups of spheres?
At the prime 2 that is.
 
 
8 hours later…
5:01 PM
Is there someone here who read the proof of Browder's theorem and is willing to help me understand the structure of the proof? From what I understand so far the game seems to be to detect the kervaire invariant 1 framed manifolds in "wu-bordism" then show they all have to be on the 2 line of the ASS then compute the 2-line of of the ASS for wu-bordism. Is that true? Even if this is true I have a very hard time identifying the arguments in Browder's paper which prove each step...
 
5:36 PM
@user40276 Related question: https://mathoverflow.net/questions/210838/does-the-forgetful-functor-from-presentable-infty-categories-to-infty-cate/210852#210852

(That this fails for $\mathrm{Sp}$ is given as a counterexample in the answer)
 
6:01 PM
@SaalHardali i don't know, but i'm willing to make a guess. the adams spectral sequences based on HF2 and on MO look the same, so i imagine "M in Omega^fr(*) has image in MO-Adams filtration n" might mean something along the lines of "there is a sequence of maps of framed manifolds M --> M1 --> ... --> Mn --> * where each Mj --> M(j+1) is nontrivial as an element of Omega^O_*(M(j+1)). that's probably not right, but i also wouldn't expect it to be far off
 
 
3 hours later…
8:41 PM
@EricPeterson Can't say I understand the reasoning. But if it is true that would be really cool.
 
 
3 hours later…
11:58 PM
@SaalHardali the stuff around 3.2.2, although stated there for complex bordism, might assemble into a less bullshit-y answer ams.org/journals/tran/2000-352-12/S0002-9947-00-02676-3/…
 

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