Homotopy Theory

3:04 AM

I think I finally "get" the additivity theorem, using the following formulation:
Thm: If $F' \to F \to F''$ is an exact sequence of functors, then $|S_\bullet(F)| = |S_\bullet(F')| + |S_\bullet(F'')|$.
Proof: Geometric realization of a simplicial space forces locality with respect to $\Lambda^1[2] \to \Delta[2]$, i.e. it "makes composites unique". In particular, $F$ and $F' \oplus F''$ are identified. QED.
So in fact, additivity already holds when you localize $S_\bullet(C)$ to become a Segal space.

I feel like this must be obvious to the 2-Segal space people, but somehow I never got the message.

Denis Nardin

7:35 AM

@TimCampion I might be missing something, but I think your proof is not complete. The key step in the proof of the additivity theorem is that F and F'⊕F'' are indeed both "composite" of the same thing. If you are not using properties special to the S-construction, you are definitely missing something

Dedalus

10:48 AM

What is the last page of the Adams spectral sequence for spheres that we know in full?

Denis Nardin

11:02 AM

@Dedalus I don't think we know in full even the $E_2$-page

And, assuming you consider $E^{t,s}=H^s(\mathcal{M}_{fg}; \omega^{\otimes t/2})$ to be "knowing in full" the $E_2$-page, I don't think we know the complete action of $d_2$ on it (unless there are some grading shenanigans I have forgotten about allowing us to reduce to computations in low degrees)

Dedalus

Thanks! That is very interesting

Dedalus

11:19 AM

@DenisNardin Is it realistic to believe that we in the near future will know the $E_2$-page in full? Are there some conjectures that would reveal much of the $E_2$-page? Of course, it is not the $E_2$-page we care about, but knowing it seems like a good first step...

Denis Nardin

I'm not going to commit, but my feeling is that understanding it significantly more than we do today is pretty much never going to happen. We might improve a bit on the differentials, but even that would be a really huge deal.

I mean, committed people might compute more groups of the E_2-page than we know now, but I don't think a "formula" for the E_2-page really exists

Peter Nelson

11:38 AM

I'm also not an expert but my understanding that the last computations of the second page were Nassau's (computer) calculations. Those are apparently almost 20 years old, so presumably those could be improved nowadays. I just don't think anyone's cared, cuz even that's way way ahead of the differentials

Denis Nardin

Yeah, maybe I should have been clearer, in that even computing the E_2-page for 100000 more values of s is not a "significant" improvement per se (in that it does not really help us to see the real "structure" of the E_2-page). Of course any technique allowing us to compute that many groups in one go would probably be amazing, but it's not really the end result that counts.

Dedalus

12:01 PM

Given that the $E_2$-page is so complicated, is there hope in finding more structure in the stable homotopy groups of spheres? Are there any conjectures which we do not know regarding its structure?

Sorry for being so vague, but I am genuinely interested and do not know too much about this.

Denis Nardin

12:12 PM

I'm very far from an expert, but I think people hope to get information on the homotopy groups of spheres from a study of the K(n)-local category (computationally we could say by attacking the chromatic spectral sequence) rather than finding those structures directly in the E_2-page of the ANSS

Tim Campion

3:48 PM

@DenisNardin To fix notation, let's say $F',F,F'': C' \to C$. Then $S_\bullet(F'), S_\bullet(F'')$ are 1-cells in $D := Fun(S_\bullet(C'),S_\bullet(S))$ from the constant functor at 0 to itself, so they give us a $\Lambda^1[2]$-horn in $D$. This horn can be filled using either $S_\bullet(F)$ or $S_\bullet(F' \oplus F'')$. By uniqueness of fillers (after localizing to get a Segal space), the result must be the same. One subtlety is that localizing $S_\bullet(C)$ to be a Segal space is enough to make $D$ be a Segal space.

I suppose the specifics of the $S_\bullet$ construction also go into recognizing that any exact sequence of functors exhibits the middle term as a composite of the end terms.

Denis Nardin

I am not sure I understand your argument. Aren't $S_\bullet(F')$ and $S_\bullet(F)$ 0-cells in that simplicial space?

Tim Campion

No, there's only one 0-cell

Remember $S_\bullet$ implicitly deloops

Denis Nardin

Also, I assume that you mean $iS_\bullet(C)$ by $S_\bullet(C)$ (otherwise you have a simplicial category)

Tim Campion

Right -- good catch!

Denis Nardin

I cannot follow exactly what you mean by "implicitely deloops"

Tim Campion

3:53 PM

Well, by that I mean that K-theory is $\Omega S_\bullet$

But the point is that $S_0(C)$ contains just the one object 0

$wS_1(C) = wC$

etc.

Denis Nardin

Yes, but isn't $Fun(S_\bullet C',S_\bullet C)$ the simplicial set whose n-simplices are maps of simplicial spaces $S_\bullet C' × [n]\to S_\bullet C$?

Tim Campion

Ok, you're right that $D$ has more than one 0-cell.

And yes, the standard meaning of $S_\bullet(F)$ etc. would be as 0-cells in $D$....

Ok

so the gap is in relating these 1-cells to the 0-cells

Denis Nardin

I mean, you're not using nearly enough. Ultimately the additivity theorem boils down to proving that some map of simplicial spaces is a realization fibration in the sense of this paper and you're not doing anything like that

Tim Campion

Do you see what I mean, though, by interpreting $S_\bullet(F)$ as a 1-cell in the functor category?

Denis Nardin

No, because it seems to be a 0-cell to me

Tim Campion

3:59 PM

I suppose I'm thinking of the case where the domain is a point.

Or rather, the domain has one object other than 0

Hm

I see how confused I am now.

Denis Nardin

The fact that $iS_\bullet C\to iS_\bullet S_1C\to iS_\bullet C$ is a fiber sequence of simplicial spaces is a triviality, the hard part of the proof is showing that it stays a fiber sequence after geometric realization. I have the feeling that you are sweeping this step under the rug

Tim Campion

I'm still guessing that because localization to Segal spaces uniqueifies composition, the additivity theorem should hold at that level, without doing full geometric realization

Denis Nardin

I mean, work with the edgewise subdivision $S^e_\bullet$ instead of $S_\bullet$. Then $iS^e_\bullet C$ is a complete Segal space already, and the above fiber sequence is still a fiber sequence of Segal spaces. You still have to prove it stays a fiber sequence after geometric realization

(the Segal space $iS^e_\bullet C$ presents the ∞-category QC by the way)

Tim Campion

If the fiber sequence splits after localizing to Segal spaces, then it will remain a split fiber sequence upon geometric realization, no?

Denis Nardin

Why so?

Tim Campion

4:08 PM

because geometric realization preserves products

I suppose when I say split, I mean "decomposes as a product"

Denis Nardin

Ah, but then you are asking a quite strong version of "split". All you know is that it has a section

Tim Campion

But this version of "split" is exactly the additivity theorem

Denis Nardin

Well, sure, but you need to prove it!

Tim Campion

Right.

But at least the classical version would follow from a Segal space version

Denis Nardin

In the proof I know it follows because it is a fiber sequence of group-like E_∞-spaces with a section, but you need to prove it is a fiber sequence

I don't think it decomposes as a product at the level of Segal spaces

Tim Campion

4:10 PM

Is that because the Q construction doesn't seem likely to split at that level?

Denis Nardin

Yeah, I don't have a counterexample at hand, but it feels just wrong

Like, assume the maximal Waldhausen structure. Then Q(S_1C) is the span category of the arrow category of C. Why should this be the square of the span category of C?

Tim Campion

although if you take the maximal Waldhausen structure, then there are a lot of extra equivalences in these categories, so really the question is if these become the same after some localization

Denis Nardin

Sorry, I meant I'm taking all arrows to be cofibrations and only the isomorphisms to be weak equivalences

Tim Campion

ok

now it makes more sense

Denis Nardin

(I always take isomorphisms to be weak equivalences when thinking about this things because there's a neat trick allowing you to always reduce to that case)

Tim Campion

4:16 PM

right

Tim Campion

11:41 PM

Ok, I think it is true after all: The additivity theorem holds after categorical realization (i.e. localizing the simplicial spaces involved to be Segal spaces), at least for input categories which are Quillen exact. Full geometric realization is not necessary. @DenisNardin 's "near-counterexample" with the Q-construction just shows that edgewise subdivision doesn't commute with Segal localization, which makes a lot of sense actually.

Let $|-|: sS_\ast \to Cat_\ast$ be the localization functor from simplicial spaces with contractible 0th space to Segal spaces with contractible 0th space. T…

Tim Campion

11:55 PM

All instances of $S_\bullet(C)$ here really mean $iS_\bullet(C)$.

And the exponentials are to be taken in the pointed categories; objects like $\Delta[n]$ should be reflected into there.

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