Dumb question. Is there any easy way to see that an E_3 structure in the nerve of a category gives a symmetric monoidal structure by using that an E_3 structure is the same as a E_1 structure in a braided monoidal category?
i have a logistical question that idk if yall have ever thought about before: i'm leaving academia, and i have a big pile of half-finished documents, project ideas, useful emails, secret documents, talk notes, course notes, blah blah blah. what's a sane way to distribute such a mess of things? i certainly can't think of any way to do that it's especially index-able, but i can't even think of a way to do it that's accessible and mildly persistent
do any of you have ideas about this, are there analogous problems that have already been solved
@LuigiM There's a lot going on in those charts. First of all, the "solid" lines of slope 1 indicate multiplication by the element $h_1$ in $Ext_{A_1}(Z/2,Z/2)$ on the Ext group described in the chart, which is the spectral sequence avatar of multiplication of the generator of $\eta\in K0_1(*)=Z/2$ in the coefficients of real $K$-theory. ...
The dotted lines of slope 1 indicate exactly the same thing, I think! The difference is in how this particular multiplication is calculated. The top 2-charts are basically describing the $KO$-theory of a cofiber $C$ of a map $A\to B$, which gives a long exact sequence $\dots \to KO_*B\to KO_*C\to KO_{*-1}A\to \dots$. The dotted line indicates an "exotic" $h_1$ multiplication, i.e., one which is not seen inside $KO_*A$ or $KO_*B$, but emerges in the extension.
In the pictures this is shown by going from blue to green.
Here's three things I thought were true: (1) The Steenrod algebra is (HF_2)^*(HF_2). (2) HF_2 is a commutative ring spectrum. (3) The Steenrod algebra is a noncommutative algebra. Clearly one of these is wrong (I'm guessing #1), but which and why?
@ArunDebray All these things are right. Why do you think one of these is wrong? (Note that the multiplication on the Steenrod algebra is given by composition, and it is not induced by the operation on HF_2)
ok, thanks. But then what's the multiplication on (HF_2)^*(HF_2)? is it the Steenrod algebra modulo some additional relations that imply commutativity?
@EricPeterson you could dump it in the Dropbox and then interested persons could peruse at will? When people look at documents they could add tags to make them more searchable in the future.
There's this paper of Devinatz and Hopkins where they give an action of Morava stabilizer group on $W[[w_1, \ldots, w_n-1]][w, w^{-1}]$ which admits a relatively simple formula and they show that there is an equivariant isomorphism $W<<u_{1}, \ldots, u_{n-1}>>[u, u^{-1}] \simeq W<<w_{1}, \ldots, w_{n-1}>>[w, w^{-1}]$ between divided power envelopes, where the right hand side is $E_{*}$ with its usual Morava stabilizer action
Are there any computations using this isomorphism somewhere in the literature?
I tried to use their formula to compute the action of the Morava stabilizer group at $n = 2$ and modulo $p$, but I struggle quite a bit.
really the ideal solution would have been to write all of it out longhand in particularly illegible handwriting and then distribute a copy to each of us by USPS
