Perhaps this question doesn't make any sense, but is the Adams tower for $H\mathbb{F}_2$ and the sphere isomorphic to the ``completion at 2 tower" for $\mathbb{S}$ in some way?

I guess I'm thinking about getting it from the Amitsur complex, and maybe we can write down the tower of partial totalizations as something like quotients of $\mathbb{S}$ by higher and higher powers of 2... But maybe we need to take $E_2$-quotients in particular?

Oh actually that brings up a possibly related question: what do you get if you take higher and higher versal $E_2$-quotients of $\mathbb{S}$ by powers of a prime, and then take the limit of that (in $E_2$-algebras)?

Haha... yeah... am I being silly by asking what $\mathbb{S}//_{E_2}4$ is?

I'm pretty sure it can be built as a Thom spectrum, at least over $\mathbb{S}_{(2)}$.

Associated to the map $\Omega^2 S^3\to BGL_1(\mathbb{S}_{(2)})$ that picks out 5...?

But then $\mathbb{S}_{(2)}//_{E_2}2^k$ has the Steenrod algebra as its homology for all $k$?

Hi!

@gexahedron I don't think the study of topological invariants of primes is developed enough to define something like the Kontsevich integral, but I might be wrong.

So far the only application I've seen is a nice topological description of some obstruction to the existence of rational points, but I'm not an expert

There's an old question I don't understand. I decided to experiment with awarding a bounty for it, but maybe I should have just asked about it here.

In the comments of the question, Jacob Lurie says that the classifying topos for $\infty$-connected objects is the localization of $Fun(FinTop,Top)$ at the co-Cech nerves of all singleton maps. I'd like to know why this is so.

And also how closely this is connected to Goodwillie calculus.

Maybe this should be obvious...

Granted this description, I can see that the $n$th object of the co-Cech nerve of the map $X \to \ast$ is the $n$-pointed cone on $X$ which appears in Goodwillie calculus, so I follow the part of Jacob's comment where he says that every polynomial functor lies in this localization. The localization is closed under limits, so functors with "everywhere-convergent Taylor towers" do too.

But I'd like to know whether Goodwillie calculus describes just part of this $\infty$-topos, or all of it.

Thanks, Denis!

Nah, I was trying to find a justification for that work I did figuring out that problem for a while :). Your main question is still very interesting though

It seems to come down to trying to commute a filtered colimit past a totalization.

The filtered colimit being the one converging to the $n$-excisive approximation, and the totalization being the localization functor you explained.

It would be neat if there were some reason these particular ones commute.

Wait I don't quite follow. Wouldn't this just prove that if F is a sheaf for the atomic topology, so is P_nF (which is true for other reasons)?

Maybe what I'm saying doesn't quite make sense. I'm thinking of a grid whose $m,n$th entry is $T_n^m(F)(X)$ where $T_n$ is the $n$th level of the Cech nerve from $X \to \ast$.

As $m$ goes to $\infty$, we get the $n$-excisive approximation, while in the other direction we get the localization, at least in the first column

Incidentally, the Čech nerve of $X\to *$ is just the cobar construction of ΣX as a comonoid (if X is nonempty, so you can choose a basepoint)

ah -- there was some discussion of this a few days ago if I recall correctly

Yeah, my question was basically whether the inclusion $\mathrm{FinSpace}→\mathrm{Space}$ is a sheaf for the atomic topology

So it's basically "yes, since the identity functor is analytic"

I see

wait

I don't really know goodwillie calculus -- does the identity tower converge everywhere?

maybe I should just read the conversation from a few days ago :)

The identity functor is only 1-analytic, so I think if this were true we would have a counterexample to your claim (but I might be getting an off by one error)

But I have since realized that the identity functor is not a sheaf for the analytic topology, because that would imply that if ΣX→ΣY is an equivalence of finite space, so is X→Y

Which I think is false

this only uses the direction that I feel confident about -- if the taylor tower converges everywhere, then the functor is a sheaf for the atomic topology.

Well, if it worked it would provide a counterexample to the other direction, but I don't think it quite works.

i see

Take $BG \to BG^+$ in nontrivial cases. The suspension is an equivalence, I think. Not finite spaces, but that doesn't matter, I think.

Well, I mean, it does matter. That's the whole point of the question, in a sense

All the counterexamples I know of the suspension detecting equivalences are highly nonfinite. Maybe there's an easy reduction I don't see but I don't think the statement above is obviously false

So how do you see that if the identity is a sheaf for the atomic topology, then $\Sigma$ reflects equivalences on finite spaces?

Well, if the identity is a sheaf for the atomic topology, then the limit of the cobar construction on ΣX is X. Now if we have a map X→Y, then it induces a map of comonoid ΣX→ΣY and if this map is an equivalence it induces an equivalence of the limits of the corresponding cobar constructions

Hm. I want to say that if $id_{FinTop}$ is an atomic sheaf, then $id_{Top}$ is an atomic sheaf, then you can repeat that argument to get the statement for infinite spaces. But I'm not sure that's any easier.

Hmm... I don't think that the first thing you said is true.

yeah, probably not.

proper subgroup

So, all subgroups of a free group are free. Isomorphism on H_1 require your subgroup to be freely generated by two elements. It seems really unlikely

Like, if you have a subgroup which induces an iso of abelianizations, but has a smaller commutator or something. Isn't there a word for group homomorphisms that induce homology isos?

@DenisNardin Your argument would also imply that $\Sigma$ is faithful, not just conservative. So a map between finite spaces which is nonzero but is zero on homology would also work. I ought to have an example on the top of my head.

what am i saying. never mind.

There exist acyclic finite spaces, see Example 2.38 in Hatcher.