Homotopy Theory

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Harry Gindi

04:39

Adeel told me that there does not exist an abelian cat of integral mixed motives

anyone know source?

If I understood him correctly, this means that the motivic stable cat does not have a heart over spec Z

only in equal characteristic

Jonathan Beardsley

05:35

@HarryGindi and @RuneHaugseng thanks for your comments. I sort of got distracted by some chromatic stuff I'm doing at the same time. But I just finished working on that, so am going to dive back into higher category theory here...

1 hour later…

Pieter

06:40

@HarryGindi Is mathoverflow.net/questions/83421/… of any help here?

Harry Gindi

06:59

@Pieter so then I heard that the standard conjectures on algebraic cycles prove the existence of a motivic t-structure. Is that only for rational coefficients?

or rather, a proof of those conjectures implies the existence of a motivic t-structure

Just bought tea; I may have gone overboard w the teabags

I used four tea bags for about a litre of tea. Is that too many?

It looks about as dark as coffee so I guess it's fine

oh yeah by the way @RuneHaugseng I said hello to you at that conference in England. So from what I gathered in you'

*in your talk, you're essentially doing a derived version of the categorical wreath product

with Δ

I was thinking it might be simpler to do it as the honest derived extension of the ordinary wreath product with Δ

If you do it like that, you should get a direct comparison with Joyal-Rezk Θ-stuff

There is a version of that wreath product where you can wreath Δ with an arbitrary symmetric monoidal category

I mentioned it to you at the end of your talk iirc

2 hours later…

Rune Haugseng

08:55

I don't really understand what you mean by that. The input to the construction I tried to describe in that talk is an E_n-monoidal infinity-category, which I viewed as a fibration over Delta^n. You could certainly do the same thing using Theta_n instead though, and you should get something equivalent.

Harry Gindi

@RuneHaugseng There is a version of Theta that produces a Grothendieck fibration over Delta from a symmetric monoidal category. Maybe I misunderstood what was going on.

I'll read through your paper and see if I can make this completely precise.

Rune Haugseng

That might be difficult, since the paper hasn't been written yet :-)

Harry Gindi

Ah, well I think the video of your talk is up on the website

if they're related, I think it is independently interesting

like, it should lead to a direct comparison functor

Could give a construction of the Cartesian-closed infty,1 cat of Theta_n infinity,n cats without model cats

I'll keep you posted if I figure it out =]

Pieter

09:14

@HarryGindi It's probably best to ask Adeel, who goes by the moniker @AAK nowadays it seems.

Too bad he doesn't get a ping though.

Harry Gindi

@Pieter I could e-mail him but he's one of my instructors and I know for a fact that he is busy rn

He's not in Regensburg; he's at a conference or something

Pieter

09:44

He might be going to the ICM then, that's the only conference happening this week, unless someone decided to compete with that.

In any case, Google brings up arxiv.org/abs/1006.1116

So it goes the other way it seems

Harry Gindi

10:06

Weird, Mitya submitted those to the Arxiv?

Is Beilinson tech-illiterate or something?

Pieter

No, see arxiv.org/abs/1505.06768

Although a quick check through earlier papers shows that he didn't submit those

Except for arxiv.org/abs/math/0204020

Interpret that as you wish :)

Harry Gindi

10:39

Maybe he learned to use the Arxiv more recently?

I dunno

2 hours later…

dhy

12:23

@HarryGindi I'm not actually really motive-literate but motivic t-structure should only be for rational coefficients ya

also i think existence of motivic t-structure should be substantially stronger than the standard conjectures but this i'm not sure about

Harry Gindi

13:18

mhm

If motives are supposed to be the universal cohomology theory, then shouldn't it have integral coefficients? How can you recover for example singular cohomology from rational motives?

Harry Gindi

13:53

like, obtaining the integral singular cohomology of the real points of a scheme

Denis Nardin

@HarryGindi My humble homotopy theorist point of view is that there's no reason why the stable category of motives (Voevodsky's DM) should be the derived category of an abelian category integrally. After all it's not true for the universal topological homology theory (i.e. the ∞-category of spectra)

You can still recover cohomology from the ∞-category of motives by taking homotopy groups of mapping spectra

Harry Gindi

Yeah, that's why I wondered what we want the t-structute for in the first place

Denis Nardin

Well the existence of the t-structure (satisfying some properties) plus the conservativity conjecture should imply all the Grothendieck standard conjectures.

Harry Gindi

can you prove the weil conjectures directly with motivic cohomology, circumventing the standard conjectures?

I mean, I know Deligne circumvented them as well, but can you follow Grothendieck's idea of proof just using motivic cohomology directly without going through Deligne's tricks?

Or do the standard conjectures have independent interest for say, the filtration of K-theory?

ah, yes they do

dhy

14:28

the main reason i care about the motivic t-structure (as an algebraic geometer) is that it gives you the bloch-beilinson filtration on chow groups

Tom Bachmann

@HarryGindi you cannot recover the singular cohomology with Z coeffcients of the real points from the motive. You can only do so with Z/2 coefficients.

Harry Gindi

@dhy Is that the coniveau filtration on K-theory that gets you back the Chow groups? We saw it in a class, but I don't remember the exact statement

@TomBachmann Ah, interesting.

@TomBachmann And you can get the DR cohomology with real coefficients as well I assume?

Tom Bachmann

14:46

@HarryGindi If X is defined over R, then its motive remembers the algebraic de rham cohomology of X, i.e. with cofficients in R. Not sure if this is your question ^^.

Harry Gindi

Yep, that was the question.

dhy

14:58

@HarryGindi No, it's different from the coniveau filtration. It's a filtration on the chow groups themselves. you can write it as a descending filtration F^i on the chow groups where F^{-1} is the entire chow group, F^0 is the homologically trivial part, F^1 is the part killed by Abel-Jacobi maps...

... and this behaves well under correspondences and which quotients F^i/F^{i+1} vanish is controlled by Hodge theory. as part of this philosophy you get Bloch's conjecture that if h^1,0=h^2,0=0 for a smooth projective surface then CH_0 is just Z

though that case now is known after ayoub

Harry Gindi

I tried reading that paper but it is beyond my level atm

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