/sci/ - Science & Math

[OHP No.8379791]

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!

[Anonymous No.8380373]

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?

[Anonymous No.8380401]

what the shit is this? is this math? how can it be math if there are no integrals? checkmate, philosophers.

[Anonymous No.8380419]

>>8380401
This is math at its best. Abstract nonsense!

[Anonymous No.8380435]

>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?

[Anonymous No.8380445]

>>8380435
>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.

[Anonymous No.8380460]

>>8379791
>Animenon-kun
absolute cancer

[Anonymous No.8380495]

>>8379791
Do you like to overhead press?

[OHP No.8380834]

>>8380435
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).

[Anonymous No.8380838]

>>8380445
Holy actual shit.

>>>/a/

[Anonymous No.8380845]

>>8380445
alice > karen

[Anonymous No.8380858]

>>8380834
Have you considered utilizing differential K-theory to translate into the language of differential cohomology?

>>8380838
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.

>>8380845
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).

[Anonymous No.8380863]

>>8380858
>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

[Anonymous No.8380877]

>>8380863
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.

[Anonymous No.8380879]

>>8380877
>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

[Anonymous No.8380893]

>>8380877

Cute! Keep doing this!

[OHP No.8381047]

>>8380879
You have nothing to contribute to the thread, and so you bash people who do? Why, anon?

[Anonymous No.8381730]

>>8381047
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.

[Anonymous No.8381738]

>>8379791
does this picture have to do with the post somehow? i'm sort of familiar with hopf fibrations but not k-theory

[Anonymous No.8381854]

>>8380435
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.)

>>8381738
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.

[Anonymous No.8381912]

>>8381738
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.

>>8381854
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.

>>8381730
Gotta go with libgen on topos theory, hypothesis confirmed.

[Anonymous No.8381969]

>>8379791
>healthy
>nice
>upstairs

stop

[Anonymous No.8381983]

kill yourself faggots

[Anonymous No.8381984]

>>8380879
this desu femme

[OHP No.8382000]

>>8381854
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.

[Anonymous No.8382582]

Anime watchers of male origin should be excluded from the genepool before they breed another generation which ends up watching even more feminine cancer

[Anonymous No.8382591]

>>8382582
>bumping a thread that hasn't had a reply in six hours just to say this

[Anonymous No.8382611]

>>8380373
>rei
>>>/trash/

[Anonymous No.8383325]

Bumping in hope for new info from OP!

>>8382591
He did like OP, he shared his insights.

>>8382611
I guess you too are my board buddy.

[Anonymous No.8383336]

>>8383325
i too am your board buddy

[Anonymous No.8383750]

>>8383336
Good morning then!