@JonBeardsley I'm around now

Does this question belong on mathoverflow?

This is the wrong room to ask.

Perhaps meta?

@Exterior this doesn't exactly answer your question, but perhaps you haven't seen the top answer to the following mathoverflow.net/questions/20740/…

@SanathDevalapurkar so K(Z/pZ,0)=Z/pZ

to get K(Z/pZ,1) we deloop Z/pZ once

in fact, we just repeat this process to obtain K(Z/pZ,n) for higher n

since Z/pZ is an abelian group, we can do this as many times as we like

if you're okay with the existence of such deloopings and stuff, then a quick computation using the loops-suspension adjunction yields the information about the homotopy groups

Meh, I forget 90% of the things I learn.

i think it's a nice picture. the original papers of eilenberg and mac lane are pretty amazing.

i did some time ago. i think they're on jstor and stuff, if you have access to that. i may have digital copies around here somewhere.

Sure.

@SaulGlasman you still around?

Saul this is more or less the thing I wanted to talk to you about:

In particular with respect to monoidal $\infty$-categories and so-called "operadic Kan extensions."

Ah, nevermind, I think that's question's bunk. I'm deleting it.

! arxiv.org/abs/1502.01713

whoa cool

Does anyone ever use tea.mathoverflow?

@SanathDevalapurkar This is not easy. It is proven in a paper of Ando-Hopkins-Rezk which you can find here:google.com/…

The relevant part is theorem 4.11

The main input to this are results of Ravenel-Wilson on the Morava K-theory of Eilenberg-MacLane spaces

@Sanath: it's unclear what one might mean by "what is K(A, n)?" i mean, as you already know, it's the unique space with pi_n = A and all other homotopy groups trivial. that tells you a lot of stuff about it already. usually it's not a familiar space, e.g. it nearly always has infinite cohomological dimension

it only looks even slightly familiar in a few cases. K(Z_2, 1) can be modeled by RP^{\infty}, and more generally K(Z_n, 1) is an infinite lens space

by a variant of kuiper's theorem, i believe K(Z_2, 2) is the projective orthogonal group of an infinite-dimensional real hilbert space

and... that's all i got

golly, this stratified homotopy hypothesis is just the bee's knees

@CraigWesterland surely one doesn't need that kind of machinery to prove statements about eilenberg mac lane spaces?

this is trying to compute the homotopy groups of certain $E_\infty$ mapping spaces

@JonBeardsley: the statement is about the space of E_\infty maps from Z/p into GL_1 E_n (that it's K(Z/p, n)).

and to map $\mathbb{Z}$ or $\mathbb{Z}/p$ into Morava E theory, yeah, there's a lot of machinery

It's meant to be amazing that the answer is (conjecturally) in fact as simple as an Eilenberg-MacLane space

Oh I guess I didn't see that part of the question.

I thought it was just about how to construct them and why their homotopy groups were trivial everywhere but one place.

(them = K(G,n)'s)

hiding in a URL :-)

I should note that as Charles says in the comments, this conjecture is true at height 1 and 2

Using the Goerss-Hopkins obstruction theory and the Ravenel-Wilson calculations, you can construct E_\infty maps K(Z, n+1) --> GL_1 E_n. Composing with Bockstein, you get an E_\infty map K(Z/p, n) ---> GL_1 E_n. I can think of this as a map of spectra \Sigma^n HZ/p --> gl_1 E_n

By adjunction, you get an S^n worth of E_\infty maps Z/p --> GL_1 E_n

this should be the fundamental class of the K(Z/p, n) that Jacob is talking about.

but why all the homotopy below it should vanish is unclear to me

Functionally, yes @SanathDevalapurkar, but it still exists.

If you find some intuition for it, let me know -- I'd love to hear it!

There's something you can do by analogy (i.e., the case n=1)

there, E_1 is p-adic K-theory, and its spectrum of units is isomorphic to BU x Z_p^x, where the BU is p-completed, and given the multiplication coming from tensor product

er, sorry, space of units

it has a copy of CP^\infty inside it, coming from the fact that line bundles are vector bundles.

here CP^\infty = K(Z, 2), but since everything is p-completed, it's really K(Z_p, 2)

that's the copy of K(Z, n+1) that Jacob is asserting exists inside of the space of units, when n=1

the construction I described above gives you a homotopy-theoretic way of producing the corresponding space inside GL_1 E_n in general

but unfortunately, it doesn't remotely have the charm of thinking of these things as line bundles sitting inside vector bundles

so: if you find a charming way to think about it, I'd love to hear it.

@Craig: a heuristic model for a cohomology theory associated to a formal group law of height n is that it should look a bit like bundles of n-vector spaces (e.g. for n = 2 this is baas-dundes-rognes and for general n this is redshift). here is in turn a heuristic definition of n-vector space: a 1-vector space is a vector space. this defines a symmetric monoidal (topological) category Vect. a 2-vector space is a Vect-module category.

inductively, if you know what n-Vect is as a symmetric monoidal (infinity) category, then an (n+1)-vector space is an (n-Vect)-module (n-category)

(although really I just want dualizable objects all around)

whatever that inductive definition means, it should in particular imply that endomorphisms of the "free module of rank 1" over n-Vect is n-Vect itself; in other words, (n-Vect)-Mod is a nonconnective delooping of n-Vect

now, the invertible objects in Vect itself can naturally be identified with BC* = B^2 Z (here I mean complex vector spaces). by iterating the above construction, there's a natural map from B^{n+1} Z to the invertible objects in n-Vect; this is the space that classifies "n-line bundles"

for n = 2 there are a couple of concrete models you can take for what a "2-line bundle" should be. there is in particular a natural map from bundles of clifford algebras to 2-line bundles given by taking some fiberwise version of the category of modules over a clifford algebra (these are, I believe, precisely the invertible Vect-modules over R or C).

in particular, start with any real vector bundle V, give it a riemannian metric, take fiberwise clifford algebras, and then complexify. the resulting bundle of complex clifford algebras has a characteristic class in H^3(-, Z) which turns out to be precisely W_3(V) = \beta w_2(V), where \beta is a bockstein

unfortunately, bundles like these only give torsion elements of H^3(-, Z), and to get the rest of them one has to do various other things, e.g. take bundles of copies of the C*-algebra K of compact operators on an infinite-dimensional hilbert space

in this case the corresponding class in H^3(-, Z) is called the dixmier-douady class, and taking global sections of the corresponding bundle of Ks is a standardish construction in C*-algebras; it classifies a particular kind of them called the continuous trace C*-algebras

@ChristopherGalias thanks, I forgot about that (great) answer. I don't know what the "right" category of measure spaces should be.

I'm a little troubled that people are calling a thing that satisfies some descent condition a sheaf – at that level of generality, every locally presentable category can be realised as a category of "sheaves"

although, I suppose in this instance, at least it's a descent condition for a Grothendieck topology

Which thing?

@QiaochuYuan It is very unclear to me that this should be called redshift. I would say it is related to redshift hopefully/conjecturally. Maybe I am wrong, but it would require knowing that iterated K-theory of $Vect_{\mathbb{C}}$ gives you $n$-vector space. Maybe this is obvious to you or others. Maybe Clark has proved this or others have. Are you saying people have proved this?

@QiaochuYuan Thanks for this. Am I right in understanding that iterating the (-)-module category construction, starting with Vect, once you pass to algebraic K-theory spectra, should produce the iterated K-theory K^n(K)? I agree that the units of this receives a map from K(Z, n+1), and that we should think of these as n-line bundles, or (n-1)-gerbes.

Ack, my indexing is off: This will receive a map from K(Z, n+2) <--> (n+1)-line bundles <--> n-gerbes

It's not obvious to me, though, that (or why) this "geometric" construction should have any higher chromatic description (going back to the original question about the units in E_n.

This was shown for n=1 by Ausoni-Rognes and Baas-Dundas-Richter-Rognes

and you can do some halfhearted things at higher n to relate the iterated algebraic K-theory to the higher "determinantal" K-theories, but they are unfortunately far from a perfect description

I guess what I'm really trying to say is: E_n is a lovely thing but I have no sense for its geometry. K^{n-1}(KU) is a vaguely geometric thing, and we have some sense that it has some chromatic level n information, especially as you say via its units (including n-line bundles). But I'd really like a more direct relationship between the two, so that we could interpret E_n in geometric terms

</rant>

@CraigWesterland Versions of character theory, dimensional reduction, q-expansions, etc for these higher geometric constructions is one reason to believe that they have to do with chromatic homotopy theory. Another reason is because they tend to have higher equivariance, and this is predicted by Ravenel-Wilson, Hopkins-Lurie, and character theory.

okay, sorry, i don't think i'm going to be able to understand this whole conversation... but... E_n is sort of geometric (in terms of algebraic geometry) in that i think in certain cases it can be obtained as the topological hochschild cohomology of K(n)

in other words, it's a deformation of a sort of spectral formal group (that is, K(n)) although this latter phrase doesn't have a meaning... yet

Further, I have reasons to believe that power operations for these geometric constructions behave just like the power operations in E-theory

@NatStapleton @JonBeardsley @QiaochuYuan I feel like I'm coming off a bit of a churlish curmudgeon here, and I don't mean to be.

Certainly not to me.

I, like you, desperately want to know what it really going on here.

That said: if you look at the Ausoni-Rognes computations, it is obvious that there is some substantial v_2-periodic phenomena going on

however, the actual answer is really kind of a mess

Right, that's what I've heard.

In particular, it's not nearly as nice an answer as the homotopy of E_n

E_n is really great.

Hahah.

You know why it's called E_n?

why?

Because Jack's wife is named Ellen.

So, perhaps you should tell me to get over myself and accept that things are never going to work out as nicely as possible, but I still hope for a description of some version of 2-vector bundles which actually recovers E_n

@JonBeardsley That's awesome

And of course it's sort of serendipitous that their associated localization functors are called L_n.

I'd be happy with anything "geometric" that recovers E_2

This reminds me of a strange conversation I had with Mahowald at some point. I was talking about the elements \eta_j that he defined, but calling them \eta_i. He corrected me, saying that they're called \eta_j to rhyme with the Kervaire elements \theta_j. Why \theta_j obviously had to end with j was lost on me.

Hahaha.

I seem to remember that $L_{E_n}$ and $L_{E(n)}$ are equivalent localization functors, can anyone confirm this and/or provide a reference?

Haha, oh my god, someone went through and starred like... everything.

@JonBeardsley I think I asked that exact question once and Eric pointed to Corollary 1.12 of Hovey's chromatic splitting conjecture paper

@QiaochuYuan !!

@QiaochuYuan @AaronMazel-Gee I'm reading it right now, it's great.

extremely cool. it took me a while to learn to relax and appreciate stratified spaces, instead of tense up and gloss over the definition

Well, they really hit the nail on the head with this conical smoothness thing. I remember being fairly unimpressed with how things worked when I was learning the Whitney story a few years ago at NadlerTalbot

Unimpressed is the wrong word. What I mean to say is that their version of stratified spaces seems to exhibit a much cleaner, intuitive theory than the Whitney theory

a related pleasing fact that david told me (which probably appears in the paper): this also -- internally to quasicategories, of course -- gives a model for "\infty-categories" which has (essentially) no preferred direction for morphisms. simply by reversing all the stratifications, or by taking "entry path \infty-categories" instead of "exit path \infty-categories", one recovers the operation of taking opposites

taking opposites of simplicial sets isn't too crazy either, of course. again it's given by a reversal of some sort

@AaronMazel-Gee Yeah, that's in the paper. That operation intertwines stratified Poincaré duality, as well, which I find extremely appealing.

ah, just got to that (only on pg 2)

It is kind of cruel that they tease a connection to the Hatcher-Waldhausen K-theory stuff but never seem to follow it up

@Sean: oh no, definitely not. i'm not saying anything about proofs. proofs are beyond me at this point :P

yeah, it seems like the stratified homotopy hypothesis should in particular give some kind of really conceptual description of intersection homology and perverse sheaves all that good stuff that my advisor cares about a lot but that i never got a good universal-property-type handle on

@CraigWesterland The "2-vector bundles" destroys the complex orientation. Or rather, taking algebraic K-theory destroys the complex orientation. The theories that are produced have higher chromatic features, but I don't think they are supposed to be natural, in the way that $E_n$ is.

@JonBeardsley So THH of an associateve and non commutative ring is a commutative $S$-algebra? I think your statement is moral and hard to rigidify to anything strict.

Jon even said THC, but I'm also surprised by the claim!

Oh interested. I think you can assemble this from results of Vigleik

Which seems to say that THH(K_n) = E_n (where K_n is the 2-periodic Morava K-theory)

so, I also have a heuristic picture of what the difference between n-vector bundles and a better behaved height-n theory is. if you believe that the stolz-teichner program can produce height-n theories out of (n, 1)-dimensional euclidean field theories over a manifold, then the difference is that n-vector bundles should at best describe n-dimensional topological field theories over a manifold; e.g. for n = 2 this is roughly the difference betwen 2d tft and 2d cft

so there are two ingredients missing, one of which is this superdirection (which there maybe needs to be more than 1 of in general; I don't understand this very well at all) and the other of which is this a priori dependence on a riemannian metric (although the supersymmetry does something funny to this), e.g. 2-vector bundles won't get you something that looks like the witten genus

although their character theory does get you something that looks like HKR character theory

but @Sean maybe all these heuristic statements will just make you madder :P

sorry, I don't actually mean "height-n theories" up there, e.g. n = 2 should produce tmf, not an elliptic cohomology theory. i mean whatever the analogue of tmf is

and also n-dimensional tfts over a manifold should be "n-vector bundles with connection," but i think the space of connections should be contractible so let's ignore that

@QiaochuYuan That would be ideal. Maybe they'll come to it in the promised sequel?

@SeanTilson @Drew this is indeed made precise by a few papers by Vigleik

@SeanTilson this isn't actually THAT surprising. there are discrete situations in which one deforms an associative thing to a commutative thing.

There is also the fact that $MU$ which is $E_\infty$ can be constructed as a colimit of $E_2$-rings.

@SeanTilson Yes, this is certainly an issue -- there's no reason that algebraic K-theory should return an oriented ring spectrum. And so that should certainly be an obstruction to relating K(KU) to E_2.

However, after localization, L_{K(2)}K(KU) is oriented with respect to K(Z, 3) in place of K(Z,2) (for the determinantal sphere in place of S^2)

this ends up making it a module spectrum for determinantal K-theory at height 2

that gadget is itself a homotopy fixed point spectrum for E_2

Though I admit that after all this, it's unclear how to return to a complex orientation in the usual sense

What do you mean by "oriented with respect to K(Z,3)"?

er, sorry, "module spectrum" can be replaced with "algebra spectrum"

@JonBeardsley Yeah, so it turns out that there is an element of Pic_n (the K(n)-local Picard group) called S<det>, and a map S<det> --> L_{K(n)} \Sigma^\infty K(Z, n+1) which behaves in many ways the same as the map S^2 --> CP^\infty

In particular, you can define a K(n)-local ring spectrum R to be n-oriented if there is a Picard-graded cohomology class x: K(Z, n+1) --> S<det> \smash R

with the property that the composite of x and S<det> --> L_{K(n)} \Sigma^\infty K(Z, n+1) being a unit

meaning:

since we can invert S<det>, do:

and you get a map S^0 --> R

which we ask to be a unit in the usual sense

Hm, ok.

Right.

With this definition in place, you can prove that n-oriented ring spectra R give rise to formal groups:

R^* K(Z, n+1) = R^*[[x]]

the group structure comes from the multiplication on K(Z, n+1)

Right.

Hm........

Ah, I should say: when n=1, S<det> = S^2, K(1)-locally

so in fact, a 1-orientation is precisely a K(1)-local complex orientation

I'm just wondering if L_{K(n)}\Sigma^\infty K(Z,n+1)-orientations correspond to.... hmmmm.... some kind of torsor condition, like.. what is the associated "MU"

Do we know the homotopy of L_{K(n)}\Sigma^\infty K(Z,n+1)?

Sort of

@JonBeardsley Sorry to go back in time a bit, but I feel like there's some subtlety here that's not being taken into account. In the discrete situation, being commutative is a condition on an associative ring, whereas being $E_\infty$ is extra data on top of being $E_1$, as you well know. Moreover, the fact that an $E_1$ ring can carry multiple inequivalent $E_\infty$ structures means that any time you get more structure out of a deformation than you're entitled to it's kind of a big deal.

Is there a determinental manifestation of the Hopf element $\eta:\Sigma\mathbb{S}\to\mathbb{S}$?

I claim that it's something like K^{det}[\psi^p], a polynomial ring on determinantal K-theory on a generator we can think of as the p'th adams operation

Yes to the Hopf map

Hm. Weird.

I wonder if being "oriented" in this sense, K(n)-locally, is sort of a one-step process. Is there a spectral sequence in which the obstructions live?

It turns out (@EricPeterson is responsible for this) that L_{K(n)} \Sigma^\infty K(Z, n+1) has a "cellular" filtration, analogous to CP^n < CP^{n+1}

Right.

Great.

there, the cofibres are (S^2)^n

here, they're S<det>^n

Fascinating.

So the "attaching map" for the new cell in the analogue of CP^2 is the determinantal \eta

Okay, so these attaching maps are the obstructions to becoming oriented in this sense?

Neat.

So, Eric and I were talking about this the other day - if you take the versal E_1-algebra of characteristic $\eta$ on $\mathbb{S}$, you get Ravenel's $X(2)$

Yeah, that's a nice way to describe the obstruction

of the sequence $\mathbb{S}\to X(2)\to\ldots MU$

ok

So I can't help wondering whether or not taking the E_1-algebra of S<det> of characteristic \eta_{det} would begin a similar sequence.

It's a nice idea

I think that Eric has some ideas on how to do the determinantal X(n)'s directly

Ah, cool.

but that would be a nice description of what the corresponding X(2) would be

That doesn't surprise me. He understands the integral ones quite well.

How does it go again? you take the Thom spectrum of the map \Omega SU(n) --> BU or something?

Yeah.

Exactly.

Yeah, I think he knows what these should be in determinantville.

nice!

@AaronRoyer fair enough. I'm mainly just saying that I'm personally no longer so surprised by this sort of thing happening.

But maybe I should be.

hey @EricPeterson

Do you guys ever see weird stuff happening like, K(n)-local Thom spectra coming from maps $X\to Pic(S(det))$ or somethiung?

OH... but I guess $Pic(S\langle det\rangle)=Pic(L_{K(n)}(S))$?

I love this quote from the recent paper of Ayala, Francis and Rozenblyum regarding the trade-off one makes in working with quasicategories: "It is technically useful to allow for coherence, but it is technically difficult to specify it."

So it's easy (and useful) to say "We call such and such a thing a _____ if it satisfies this enormous prepackaged collection of coherence data."

But much harder to take an arbitrary object and prove that it is such a thing.

"j" in "theta_j" presumably because of the relation to h_j^2, or h_{1j}^2 with the i index suppressed

i don't know peter's wife's name, but i'd be surprised if it fit an "h i j" template & this was a broader phenomenon :P

at some point in the past i did think i knew how to build X(n)^det, but i don't have a lot of faith in past-Eric now & i've been avoiding going back to actually look and see if it's salvageable. who knows

@JonBeardsley & lastly i think pic should be applied to ring spectra (or module categories), but S<det> isn't (known to be) one of those. certainly craig's M^det (or MX or whatever it's being called) does come from thomifying a stable bundle of K(n)-local spheres

& past lastly, i liked this folk etymology better chat.stackexchange.com/transcript/9417?m=12640522#12640522