Looking to share a new insight in particular with Animenon-kun regarding spectral sequences from the nPOV.
Given some cohesive infinity topos of spaces, we have in particular Cartesian closed structure given by the product-hom adjunction. Also, assume we have a healthy notion of spectral sequences for our spaces. Then, given a nice fibre sequence [math] F\hookrightarrow E\rightarrow B [/math] upstairs, we have a spectral sequence [math] \pi_{0}(A^{F})^{B} \Rightarrow \pi_{0}(A^{E}) [/math] for a coefficient object [math] A [/math]. Now, reducing to the first K-theory, this ends up being the isomorphism [math] ([A]^{[F]})^{[B]} \simeq [A]^{[F]\times [B]} \simeq [A]^{[E]} [/math], natural in [math] [A] [/math], where the second equivalence comes from the K-theory identity [math] [E] \simeq [F] \times [B] [/math]. (I have used multiplicative language for this K-group since I feel that it is more "correct;" typically one uses additive language, but that clearly does not fit. The choice is technically arbitrary, but it ought to remind us of the higher stuff going on.)
So, from the nPOV, spectral sequences are a homotopy-coherentized consequence of higher Cartesian closedness. This will be a remarkable fact once I flesh it out more, since it means that we can use spectral sequences in any infinity topos! In particular, it shows why spectral sequences are able to be lifted to stable- and equivariant-homotopic settings, as these both inherit cohesive structure from their base topos. I would like to formulate this in terms of differential cohomology theory!
Ethan Clark
Right, let's see. I must start with an apologetic comment that I know next to nothing about K-theory and differential cohomology or topoi/toposes/whatever, but I'll bump to keep the thread alive.
This sure looks interesting, and I need to read a bit more about this stuff to comment properly, but I somewhat see your point. If it really is the case that we can have spectral sequences in an [math]\infty[/math]-topos, does this help you reach your goals regarding the stuff you mentioned yesterday?
Connor Hughes
what the shit is this? is this math? how can it be math if there are no integrals? checkmate, philosophers.
Henry Gonzalez
This is math at its best. Abstract nonsense!
Thomas Evans
>cohesive infinity topos of spaces If it's cohesive even, I guess we already consider the objects space, no?
I also don't know K-theory. What's [·] so that the bundle gives rise to the "topologically simple" relation [E]≃[F]×[B] ?
You claim there is a misnomer with + and ×? What about the cardinalities of those things after applying the forgetful functor, has the + object the sum or the product number of elements?
I don't know the nPOV of spectral sequences because I don't have any other POV. I'm sure I could have invented them, but plz tell my why they are interesting to study. Is it more than classifying spaces?
And if you conclude that they exist in any topos, could you plx write down the simplest topos (is FINSet one?) and tell me what they are there?
Who is Animenon-kun, what does he do?
Adam Campbell
>Who is Animenon-kun, what does he do? It is me, O(H)P's name for his boardbuddy. What I do seems to be mostly that I give him encouragements and receive interesting info from OHP. In addition, I study this kind of stuff but at a significantly lower level.
Oliver Price
>Animenon-kun absolute cancer
Alexander Hall
Do you like to overhead press?
Oliver Ramirez
Check out the nLab article on K-theory; the reason for the misnomer (it really isn't one) is that it comes from algebraic K-theory, and dealing with opposite categories means we replace those products with tensors (between projective varieties).
Jack Edwards
Holy actual shit.
Julian Jackson
alice > karen
Wyatt Davis
Have you considered utilizing differential K-theory to translate into the language of differential cohomology?
That's just rude. I'm getting barraged with these kind of messages. This must be what a terminal object feels when all arrows point at it, nothing.
I think so too. Alice is shy, and thus easy to identify with in some sense (not that I was crazy enough to think I'm a cartoon girl).
Daniel Scott
>I'm getting barraged with these kind of messages. You know why? Because trying to develop an identity for yourself on an anonymous science board as "that guy who includes an irrelevant anime head with every post" is fucking gay
Nathaniel Ross
Suppose that is my intention. Then posting angry messages would just give me more fuel, wouldn't it? So, in other words, you are making the situation worse by doing this. Nevertheless, this is irrelevant. This thread (which I was literally invited into) is about OP's insights reached by studying spectral sequences and K-theory, not about my motives. Thus, I shall derail this thread no more by answering to fan mail.
Jason Clark
>So, in other words, you are making the situation worse by doing this. It's not like you're going to go away anyway. And it's only possible to "invite you" in because you're avatarfagging and using this board like a skype chat in the first place.
tl;dr commit suicide you pathetic lonely cunt
Caleb Martinez
Cute! Keep doing this!
Owen Ross
You have nothing to contribute to the thread, and so you bash people who do? Why, user?
Lucas Martin
Just ignore them. I think it's better that they let their steam out here instead of beating up (or getting beaten up by) a guy on the streets. They bark, but they won't bite.
However, I started reading Atiyah's book on K-theory last night. Since I almost fell asleep because it was 5 am, I guess I'll have to re-read the stuff I got through, but rehearsal is the mother of learning! I'm about to visit my uni soon, and shall check if there's a book on topos theory in the library (which there probably isn't because it's basically all ANALysis). Maybe in a few days I can really contribute something.
Dylan Jackson
does this picture have to do with the post somehow? i'm sort of familiar with hopf fibrations but not k-theory
Ryan Sullivan
I guess you respond to the \plus-\times issue, right?
Okay, what about the applications, is it about classifying topological spaces? Or is it the road to the solution to some "sensible" problem. (Where I mean some problem that didn't arise only after the concepts have been set up.)
I guess it has to do with F↪E→B and that suffices. The likely alternative to make it thread relevant would have been a grid depicting a spectral sequence.
Lucas Thomas
The choice of image is there for three reasons: >a file is needed to start a thread (OHP doesn't use images) >it is an example of a fibration >the image looks nice I suppose this is pretty much it.
Yes, the notational thing is about [math][E] \cong [F] \times [B][/math], where there "should" be [math][E] \cong [F] + [B][/math]. That's the thing he meant, but (as far as I understood this thing with quick checks from Wikipedia and nLab) there's no problem with his notation.
Gotta go with libgen on topos theory, hypothesis confirmed.
Cooper Howard
>healthy >nice >upstairs
stop
Zachary Hernandez
kill yourself faggots
Jordan Lewis
this desu femme
Caleb Ward
Regarding applications, it may end up being useful down the line for classifying principle bundles over spaces, and in that vein it probably has applications within higher Chern-Weil theory. So, it has physics applications as well. I'm just excited because it helps me understand things better now! I had no idea beforehand that spectral sequences were related to Cartesian closedness. Just that illumination is enough to make me want to share it.
Zachary Bennett
Anime watchers of male origin should be excluded from the genepool before they breed another generation which ends up watching even more feminine cancer
Samuel Foster
>bumping a thread that hasn't had a reply in six hours just to say this
