Homotopy Theory

« first day (1968 days earlier) ← previous day next day → last day (1435 days later) »

12:04 AM

@OmarAntolín-Camarena Also isn't there a functor (N with many objects) -> (N with one object) pulling back along which produces the thing that I want?

@TylerLawson thanks, that's very helpful. I did exactly what you said in the first paragraphs about producing the multiplication, just to see what goes it. Didn't have time to work out more yet (or rather I thought that there surely must be a reference... maybe not)

skd

does anyone have a copy of "Mappings of quaternionic projective spaces" by Feder and Gitler?

Omar Antolín-Camarena

12:32 AM

@TomBachmann Yes, that sounds right. The reason I like the left Kan extension version, is that it explains why the telescope inverts a (and that it does so in an initial way).

Marc Hoyois

12:52 AM

@OmarAntolín-Camarena However, the automorphism of E[1/a] you get from this left Kan extension perspective is not necessarily homotopic to the action of a on E[1/a] (coming from the E-module structure on the latter). That's been a source of confusion for me in the past...

Harry Gindi

1:30 AM

@DenisNardin Well Danny Stevenson's new paper would help greatly.. The other generating cofibrations are not too crazy

9 hours later…

Saal Hardali

10:03 AM

The category of modules over a commutative algebra has an extra structure (given by tensor product) which is formalized by the "symmetric monoidal" structure which makes sense on a general category. Similarly the category of comodules over a cocommutative coalgebra has an extra structure of cotensor product of comodules which is suspect is not a symmetric monoidal structure. What kind of abstract structure is this?

Saal Hardali

10:21 AM

Is it just a symmetric monoidal structure on the opposite category?

Harry Gindi

11:20 AM

a symm monoidal str on the opposite category seems like it just gives a symm monoidal str on the original one, doesn't it?

all of the structure maps and coherences are reversible

I'm thinking maybe stupidly that it's symm. monoidal closed on the opposite category?

so that it's somehow right adjoint to the internal hom instead of left

that way you get some non-reversibility into the mix

Fernando Muro

the cotensor product is often not associative

the problem is that it is defined by means of tensor products and kernels

and they seldom get on well

but when they do, under flatness assumptions, the cotensor product is a monoidal structure

Harry Gindi

If it's associative under a flatness assumption, does that mean that there is an associative derived version @FernandoMuro

1 hour later…

Saal Hardali

12:35 PM

At least when the coalgebra $C$ is finite dimensional over a field and the comodules are finite then the cotensor product is equivalently described as taking linear duals, tensoring them over the dual algebra $C^{\vee}$ then taking duals again.This is why I thought it may be an op symmetric monoidal structure.
In any case a commutative algebra can be reconstructed from its symmetric monoidal category of modules I would like to have a version of this for the category of comodules over a cocommutative coalgebra + some additional structure (which is think ougt to be the cotensor product).

2 hours later…

Fernando Muro

2:51 PM

@HarryGindi the cotensor product looks like the H^0 of a bar-like construction, but I don't know how associative is that, probably not much as you'd likely recover coassociativity of the cotensor product by taking H^0. but I'm not sure, all I'm saying right know may be nonsense

Harry Gindi

3:15 PM

@FernandoMuro Just wondering because in a lot of DAG situations, you can remove flatness assumptions by working with derived functors the whole way through. This may be unrelated entirely though!

Saal Hardali

@FernandoMuro Maybe its worth pointing out at this point that what i really care about is the stable category of complexes of comodules with cotensor product the two sided cobar construction. Any classical statement should be possible to recover from this if it behaves well enough w.r.t. t-structure

Its also worth pointing out that in the calssical case without some assumptions (e.g. flatness over) the base-ring the category of comodules is not even abelian so really the reasonable thing to do is to go derived from the start in my opinion.

3 hours later…

Piotr Pstrągowski

5:51 PM

@HarryGindi In the abelian case, the cotensor product is given by an equalizer which is a finite limit, and so under flatness assumptions you can commute it with a tensor. However, my guess would be that in the derived case the cotensor product will be given by the a totalization of a cosimplicial object, and these are not necessarily preserved even by derived tensor products since it's not a finite limit.

Note that I'm not saying that the cotensor product cannot be associative, just giving an argument why passing to the derived setting doesn't necessarily solve the problem.

Saal Hardali

6:15 PM

@PiotrPstrągowski Good point, Thanks! Totally unrelated but I was just reading your paper "chromatic homotopy is algebraic for p>n^2+n+1" (which is absolutely awesome btw) and I was wondering (I hope its okay to ask): in the notation of the paper you show that $\beta_*\mathbb{1}_{\le l} \cong P_{\le l}$ as associtive algebras. Do you have evidence suggesting that they are inequivalent as commutative algebras? Or is it just the case that it seems difficult to prove?

2 hours later…

skd

8:36 PM

the stable element alpha_1 actually comes from an element of pi_{2p}(S^3). does alpha_t come from an element of pi_*(S^3) of degree 2t(p-1) + 2?

Ryan Keleti

8:52 PM

hello all, probably naive question but in practice which model/presentation for the stable homotopy category is used? Via EKMM we have the category of $S$-modules, and via Hovey-Shipley-Smith the category $Sp^\Sigma$ of symmetric spectra. These seem to be the main constructions (?), but we can also talk about spectrum objects in a more general sense via stabilization of an $\infty$-category. Any pointers or sources would be greatly appreciated!

Denis Nardin

@RyanKeleti In most of the works it doesn't really matter. I personally always use the ∞-categorical approach or, if I really need to choose a model, orthogonal spectra, but other people have other preferences

In practice most papers can be easily rewritten from an approach to another, with only very few exceptions

Ryan Keleti

we can talk about Quillen equivalences between these different models, but I'm not familiar with how one passes from one model to the next

Denis Nardin

Well, ideally the statements should be model invariant

But more concretely I was thinking of "an expert can easily rewrite the proof with another model if they so prefer"

Ryan Keleti

that makes sense, I should read up more on the model structure of the individual constructions

Denis Nardin

Nah, you shouldn't

Ryan Keleti

8:57 PM

ok! but I wonder if one has to be somewhat 'proficient' in each theory

skd

(this reminded me that mark hovey wrote in his problems page: "I am not sure working on model categories is a safe thing to do. It is too abstract for many people, including many of the people who will be deciding whether to hire or promote you. So maybe you should save [these problems] until you have tenure.")

Denis Nardin

If you ask my personal preference (and again, I stress that other people might disagree), writing the whole paper in a "model independent" way is typically quite easy. I tend to use the ∞-categories both because of this (they tend to encourge you to write without paying attention to the model) and because of the added flexibility that it provides me

The fine details of the model are really not the things you should prioritize in learning homotopy theory. There's time enough later if they become important in your work

Ryan Keleti

the $\infty$-cat approach seems the most appealing right now, but I don't know enough as to whether the other constructions are more useful in certain situations

they certainly feel more 'concrete,' if that means anything

Denis Nardin

If you really want to learn a model, I recommend orthogonal spectra because it's the simplest. But in practice it won't matter much

Ryan Keleti

I'll take a look, thanks for your help!

1 hour later…

Saal Hardali

10:14 PM

Let $\mathcal{S}_{\mathbb{Q}}$ be the initial $\infty$-category among those factoring the rational stabilization functor $H\mathbb{Q} \otimes \Sigma^{\infty}(-): \mathcal{S}^{\ge 1}_{\ast} \to Sp_{\mathbb{Q}}$ (rational localization of spaces). Is $\mathcal{S}_{\mathbb{Q}}$ equivalent to the $\infty$-category of rational dg-lie algebras?

I feel like this should be true but everywhere I look I always find the restrictive assumption of simply connectedness.

Harry Gindi

10:29 PM

@RyanKeleti Spectra are one of the specufic cases where working model-independent has a completely explicit theory in infty-cats that transcends model cats

the equivariant versions still need to use model categories though, iirc

« first day (1968 days earlier) ← previous day next day → last day (1435 days later) »

all rooms

Transcript for

Nov16

Nov '1817

Nov18

A room for anyone interested in homotopy theory, or any nearby...

homotopy-theory

join this room
about this room

00:00

06:00

12:00

18:00

all times are UTC