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6:35 AM
An event is scheduled here by me on the occasion of Karl Weierstrass' birthday. Please register if you are interested.
 
 
6 hours later…
12:42 PM
@AaronMazel-Gee The book in question is "handbook of model categories" which I just finished writing and is with referees at the moment. I guess it is kind of like "counter examples in topology" but model categories
 
1:32 PM
@ScottBalchin Sounds great! I look forward to reading it when it becomes available!
 
 
1 hour later…
2:37 PM
Ironic that all these texts are called 'handbooks' when (e.g.) the Handbook of Homotopy Theory weighs 4.16 pounds and is nearly 1000 pages long
But I suppose it's text to have 'at hand' rather than 'in your hand'
 
None of them is a catalogue of hands, so once that band-aid is ripped off I think anything is possible
 
 
8 hours later…
11:02 PM
Is there a formal-group-theoretic significance to the fact that the generator of the degree zero mod 2 cohomology of the Brown-Peterson spectrum BP (at p=2) exhibits the mod 2 homology of BP as the subalgebra of the dual Steenrod algebra generated by the squares of the usual generators?
 
11:47 PM
There's probably deeper things that can be said, but one thing to notice here is that $H_{\bullet} BP$ (which is a subalgebra of the dual Steenrod algebra $H_{\bullet}H$) is isomorphic
$H_{\bullet} \otimes _{BP} BP_{\bullet} BP \otimes _{BP} H_{\bullet}$, which is the Hopf algebroid (in this case, even a Hopf algebra) corresponding to the automorphisms of the additive formal group, namely the formal group $Spf(H^{\bullet}(CP^{\infty}))$.
This is the more fundamental fact in the sense that this holds at all primes, not just at $p=2$.
From this perspective, I think it is a little bit more interesting to ask what is the significance of elements of $H_{\bullet} H$ that do not live in $H_{\bullet} HP$, as these do not a priori have interpretation as anything coming from formal groups, or in any case from the Quillen formal group $H^{\bullet} CP^{\infty}$.
At $p = 2$, an interesting thing happens, because instead of $H^{\bullet}(CP^{\infty})$ you can consider the formal group law on $H^{\bullet}(RP^{\infty})$. These two are in fact isomorphic if we disregard the grading (in terms of coefficients rings, through the Frobenius), and similarly $H_{\bullet}BP$ is isomorphic to $H_{*}H$, also through the Frobenius.
This way, you can identify $H_{*}H$ with representing automorphisms of $H^{\bullet}(RP^{\infty})$, but I would say that this is a consequence of the same fact for $H_{\bullet} BP$, which is the more fundamental object.
(A similar story plays out at odd primes, $H_{\bullet} H$ corepresents automorphisms [in an appropriate sense] of $Spf(H^{\bullet}(BC^{p}))$. It is, however, no longer true that $H^{\bullet}(BC^{p})$ and $H^{\bullet}(CP^{\infty})$ are isomorphic, even after disregarding the grading.)
 

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