oh snap!~

@Drew thanks brue!

asked that question from last night on MO: mathoverflow.net/questions/138132/…

looks awesome.

btw @EricPeterson love how u give background on ur MO questions, that's rare

always be complete

p-adically and also in writing

hehe, yes to the latter, not to the former

what is $X[n,\infty)$ for a spectrum X? @EricPeterson

oh maybe ur saying here, connected cover?

yeah, what some people call $\tau_{\ge n} X$

the thing with X's homotopy starting in dimension n

i really like the interval notation

@EricPeterson: nice question.

I am curious, and I think people have alluded to this before (or perhaps there is a paper I know of where this comes up), how far is BGL_1 from being suspension?

Sorry, it is never going to be the honest suspension. Also, what is the reference for that formula where you compute GL_1(R)? or perhaps it is obvious.

I guess it is supposed to be obvious from the description as a (homotopy) pullback.

@ChristianNassau: I was looking at your website and I noticed Yacop. I am curious if you really need hopf algebras. Could one get away with just bialgebras?

yeah exactly, it's obvious from the pullback definition

idk about the suspension question

(Shameless self-promotion: ms.unimelb.edu.au/~dheard)

Finally made a website

And got it working. That was hard

And stole @JonBeardsley's links list. So if you're not on it, blame Jon

@Drew i can't see your icons at the bottom of the page, they appear as broken images

Oh thanks. They work for me, but I might need to change permissions

Oh that's because I'm looking at my local site. Opps

Should be fixed now

looks good :)

Great, thanks for the 'beta'-test

@Eric Yacop is actually only deals with the Steenrod algebra and its sub Hopf algebras - these routines are hardcoded. Also, the Hopf structure is not really used for the computation of a minimal resolution - but I don't know any interesting subalgebras of the Steenrod algebra that are not also sub Hopf algebras.

you mean to highlight @SeanTilson

but: hi

Thanks @Eric. And also Hi!

@SeanTilson : The coproduct does become important when you compute ${\mathrm Ext}(M,N)$ for general $M$, $N$: here I'm using $C_\ast \land M$ as resolution of $M$ and the fact that anything smashed $A$ becomes free needs the Hopf structure.

@Drew gorgeous site.

and of course, my links are available to everyone

:)

@EricPeterson dang.... NO responses to ur question

yeah :( but it's been less than a day, and also tyler hasn't been around here recently, which may mean he's not around there either, and he's who typically answers my questions

haha true.

I was about to comment, but I wasn't sure whether my comment addressed the question.

What came to my mind is that your spherical fibration has a complement $\psi$, and from $T(\phi) \land T(\psi) = \Sigma^nX$ you can recover $X$

that's true

So no information is lost when you go from $X$ to $T(\phi)$. But that might not have been the question...

Somehow, speaking of "attachments" is a convenient mnemonic (that I use often myself), but it's hard to make presice.

right, the question was to understand how the information gets sheared. maybe it's useful to note that it can be sheared back, though, idk

Well, I did like your question! I'm not sure that helps, though ;-)

encouragement is always nice

what do u mean by sheared

like, spread out across the spectral sequence?

wow.... what i'm trying to do for group schemes has already been done (EXACTLY THE WAY I WANT TO DO IT) for "abelian group functors"

i'm simultaneously ecstatic and bummed, lol

so if the baby question is true, for instance, then the rough formula is something like 0 (attaching map of the n-cell in S^n) + omega (action of the spherical bundle) = omega (attaching map in the thom complex T(phi)). maps that look like (x, y) |-> (x+y, y) are called shears, and an important feature is that they're invertible (that one by (x, y) |-> (x-y, y), and the previous one by picking the spherical bundle -phi)

ahah, yeah, okay, i was wondering about those shearing maps

nice little dudes

i'm strictly enthused when i find out someone's already done something i want. i often accidentally ask hard questions, and it's great news that someone else has already struggled through answering it and i don't have to

work sux

hahah yeah ultimately i agree with you

now i can just say "from work of Breen, we know...."

it also let's me know i'm on the right track.

thank goodness for larry breen

breen even uses the spectral sequence of Quillen that Quillen writes down in (co)homology of commutative rings!

god i'm so excited

oh larry breen you dog. you don't even know.

i really should have just done algebra/category theory.

it's not too late, you have your whole life ahead of you

that's true. and more and more it's looking like my thesis is going to be heavily invested in this stuff. i feel like i'm finding that there are a lot of sort of... forgotten algebraic formulations that could really easily be generalized to "brave new algebra" but haven't been

maybe it's because they're not useful, but having this re-proof of lazard's theorem in my back pocket at least provides one apparently meaningful application

oh my god, i really need to talk to this guy. he wrote a paper on "classifications of 2-jets and 2-stacks"

oh you're kidding me... and another one called "Extensions of the additive group"

i would like to know what breen's Sigma-structures are

but not today

lol

what is that related to?

Theta-structures, biextensions, the theorem of the cube

oh

it's chapter 5 of fonctions theta et la theoreme du cube. that monograph is a pretty deep rabbit hole, i've barely read any of it

i see

oh good, tyler is around after all

@JonBeardsley wait, which paper of Breen has this fancy proof of Lazard's stuff? I wanna read it!

not a proof of lazard's theorem

a proof of a theorem of lazard's, haha,

@DylanWilson it's the theorem that an n-bud is extendible if and only if $\Gamma_n(f)$ is a coboundary in a certain group scheme homology

or maybe more like the cohomology of the global sections

and it's not proven in anything breen does, he just seems to give a useful framework for it

sciencedirect.com/science/article/pii/S0022404998001406

hey @aaron u there

@ChristianNassau I am interested in computations involving the Dyer-Lashof algebra, so far it sounds like your software would be potentially adaptable. The Dyer-Lashof algebra does not have an antipode, but it is superficially very similar to the steenrod algebra. Have you thought about this at all?