@lentic: spectrum <-> abelian group is an analogy. spectrum <- chain complex is an inclusion; chain complexes present nice spectra

@QiaochuYuan you mean because of $\mathcal{D}(R)=\textrm{Mod}_{HR}$ ?

In view of what has been written above, the analogy seems clearer to me if we consider chain complexes instead of just abelian groups. Why do some people usually take abelian groups as the "algebraic analogue", anyway? Is it because $\Omega$-spectrum "=" grouplike $E_\infty$-space which feels similar to an abelian group?

isn't that only for connective spectra?

yes

also, something I'm not clear on: is the forgetful functor from D(Ab) to Sp fully faithful?

no it's not

There are no maps from Z to Z[n] in D(Ab) when n is not 0

but there are plenty of interesting map from HZ to HZ[n]

right, that's what I heard... so it's misleading to think of chain complexes as being special spectra

Agreed, they should be thought of as spectra with additional structure.

i would say chain complexes of abelian groups are more directly analogous to abelian groups, while spectra are a nonlinear analogue of abelian groups

@GeoffroyHorel the maps HZ ---> HZ[n] you mention they are just maps of spectra, right? not of HZ-modules, correct?

that's right

so the morava stabilizer group $S_n$ is a profinite group. a profinite group is isomorphic to the inverse limit of a system of discrete finite groups.

what are the finite groups $G_i$ whose inverse limit is $S_n$?

@Josh Its quotients? I do not think they have a particularly nice intepretation. $S_n$ is usually identified with the group of units of a complete algebra. For example $S_1=\mathbb{Z}_p^\times$

Z_p is the p-adic integers, right?

Yep

ok

what's S_2?

Ugh I do not remember the algebra in that case.. It's somewhere in Lurie's notes on chromatic homotopy theory

@DenisNardin

ok thanks

what do you mean by quotients?

If you have a profinite group $G$ it has a canonical description as a direct limit of finite groups. Take the lattice of open subgroups $\{U\}$ and send $U\mapsto G/U$

so G is isomorphic to lim_U G/U

right?

yes

Uh wait, you want normal open subgroups

ok that's cool

one more question

if n is an integer and $\hat{Z}$ is the profinite completion of $Z$ the integers then is $n\hat{Z}$ finite?

i think the answer is no

Uh, certainly not. $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$

right so it contains $nZ$ which isn't finite

correct?

Among a lot of other things it contains, yes :)

great, thanks

i'm a little shaky on this stuff

i don't know chromatic homotopy theory well and i'm trying to learn it

thanks a lot

What's your background?

It may be fruitful to try to get a solid base before running into the advanced stuff

mhm

it's that i've studied higher category theory via simplicial sets

but not chromatic homotopy theory

Yeah, ok but higher category theory is just a thing. Do you know classical homotopy theory? At least the basics of homology and cohomology? Topological K-theory? Otherwise a lot of this stuff may seem unmotivated

yeah i know that stuff

And also a smattering of algebraic geometry and number theory really helps

mhm that's what i lack

i'm perfectly comfortable with anything topological

but i know a moderate amount of algebraic geometry and nothing much about number theory

Well, good luck then :)

thanks my friend!

@lentic: a grouplike E_{\infty} space is not just similar to an abelian group! abelian groups are precisely grouplike E_{\infty} spaces whose underlying space is discrete

the analogy should really go spectrum : abelian group :: space : set which I think might make you a little happier

abelian groups are 0-truncated and 0-cotruncated spectra, in the same way that sets are 0-truncated spaces

@Qiaochu Well abelian groups are also 0-truncated and cotruncated elements of $D(Z)$ :)

yeah, again the correct analogy is really with D(Z)

And Spt is the nonlinear version

Spaces : Some Symmetric Monoidal Category Of Spectra : Homotopy Category Of Spectra :: Sets : Chain Complexes Of Abelian Groups : Derived Category Of Z

Eilenberg-Mac Lane Spectrum :: Abelian Group

would you agree?

@Adeel I really dislike calling spectra "nonlinear". I understand that you mean that they have no $H\mathbb{Z}$-action but it just rubs me wrong

I'm with Denis here

I don't understand why, but OK

@lentic More or less, with a few caveat that I usually think in terms of stable categories rather than homotopy categories

spectra are so linear that you can add and subtract morphisms

there are zero morphisms

how could you get more linear than that?

maybe "nonabelian" would be better?

they have direct sums

I mean that is what I would call additive

if by linear you mean over a field, then abelian groups aren't linear either

I don't know a way in which abelian groups are "more linear" than spectra

Also I actually changed my mind about the analogy, here is another version: sets : abelian groups : R-modules :: spaces : spectra : HR-modules (= chain complexes of R-modules)

where you have to forget that abelian groups = Z-modules

@Saul I definitely don't agree, but I have to go

I think that people are unconfortable because spectra quack like a derived category, walk like a derived category but are not a derived category

I think we should embrace it and think of them like a derived category of a nonexistent ring

@DenisNardin I think the discomfort comes more from the lack of a concrete description. The bounded derived category of a ring is just the chain homotopy category of projectives. That's pretty concrete, you could teach it to anyone who's seen what a module is.

Hmm... I don't know, I have the impression that this is more a didactic failure than anything else. I personally find the description of spectra as pointed 1-excisive functors (aka space-valued homology theories) quite intuitive for anyone who has seen what homology is

@DenisNardin I'm not familiar with that description, could you expand on that?

Expecially if you follow by the descriptions of Eilenberg-Maclane spectra and connective ko in this language

I will use "finite space" to mean "space homotopy equivalent to a finite CW complex" and "functor" to mean "functor that is continous on mapping spaces".

So you can describe homology theories as follows: an homology theory is a functor from finite pointed spaces to abelian groups such that $H(*)=0$ and for every cofiber sequence $X'\to X\to X''$ the sequence $H(X')\to H(X)\to H(X'')$ is exact

You can now define a spectrum as a functor $E$ from finite pointed spaces to spaces such that $E(*)=*$ and for every cofiber sequence $X'\to X\to X''$ the sequence $E(X')\to E(X)\to E(X'')$ is a fiber sequence

You can then easily prove that a spectrum is the same thing as an infinite loop space

hmm, by functor to you mean actual classical functor?

Well yes, with the caveat that the map $\Map(X,Y)\to \Map(E(X),E(Y))$ must be continous

I'm secretly using topological categories as model for $\infty$-categories but you do not need to tell

that's actually neat.

do you have a reference for that?

and you don't get into trouble if you try composing morphisms or with homotopy equivalences? I'm not sure about which technicalities I'm worried about.

@pro Everything works fine but you need to be a little careful. You need to endow the maps between spectra with an appropriate topology but then an homotopy is just a path between them

(there might be point-set problems if your category of spaces is not good enough but I consistently ignore it)

@lentic Unfortunately none that would be accessible for a beginner. Maybe Goodwillie Calculus' papers?

so this gives a model for connective spectra? is it easy, for example, to show you get a triangulated category?

No, this gives a model for all spectra

And yes, the loopspace functor is applied pointwise and you need to verify that it is an equivalence (in the topological category sense) but that's not hard

Fiber sequence are pointwise too

Oh and the definition of $H\mathbb{Z}$ is amazing. It is just the functor sending a space $X$ on the free topological abelian group $\mathbb{Z}X$

(All of this is in fact much easier to do with simplicial sets but if you're determined enough you can do it with topological spaces too)

oh, come on, this should be way more popular

I wholeheartedly agree

agreed

you should write a note explaining this with more details!

whenever I've heard people ask the question "so, is the cat of spectra the same as the category of cohomology theories? (interpreted as functors from the homotopy cat)" the answer has always been "absolutely not, you need to operads, loop space, etc etc etc"

a star to that message in the hopes it will get more, and it gets @DenisNardin to succumb to peer pressure and write that note

hah

I already tried to give a general (math) audience talk on this and it ended in a bloodbath

I'm not sure I'm the right person for this job

well, next time don't carry an axe with you

that axe hurt the general math audience but it wouldn't hurt (apprentice) homotopy theorists, wouldn't it?

At least the sympletic geometers in the room appreciated the talk. People in applied math a lot less...

as a different sort of topologist I'd also love to see the details of the above idea written down somewhere

Oh and let me tell you how you define the smash product here

You just say that a bilinear map of spectra $E\times E'\to F$ is just a natural transformation of functors $E(-)\wedge E'(-)\to F(-\wedge -)$ and then the smash product is the universal bilinear map

Just like the tensor product of abelian groups

(For those not in the known this is basically the localization of the Glasman-Day convolution)

does anyone know what does Cartan-Eilenberg mean when they talk of a "group with unit augmentation"? I think they only say it once, in p. 195, for doing stuff with its group algebra

@Adeel I'm with Saul and Denis also. If you linearize something, you get something linear!

@lentic I think that "with unit augmentation" refers to the word "ring" and not to "group"

damn, yes, that makes much more sense. They should definitely have used a comma there

question for the room, somewhat related to my last question

suppose I have a map of simplicial sets f : X -> Y such that Y is weakly contractible and so are all the fibers of f

if I want to conclude that X is contractible, what's the least hypothesis I can get away with on f?

the most natural answer is Kan fibration, but actually you don't need that - cocartesian or cartesian will do it

can I do any better than that?

@Saul I'd say to use Quillen Theorem A (that is that the hypothesis is that the map is cofinal).

A cocartesian fibration with contractible fibers should be cofinal and so this should be a generalization

ah I guess so

or maybe a cartesian fibration with contractible fibers is cofinal and the cocartesian one is coinitial?

either way

Maybe I can never keep them straight

I really want there to be an op-invariant condition, but maybe life just isn't that good

so you mean strict fibres?

yeah, point-set fibers

Regarding the linear thing: I mean, spectra are linear over the sphere spectrum. Abelian groups are Z-linear. If linear means S-linear for you, well ok, but you have to admit this is much much weaker than the classical use of the word

my semantic intuition is that "linear over something" should imply "linear"

Maybe it's my homological algebra background then

@SaulGlasman well, then the most natural condition is that homotopy fibre = strict fibre, i.e. quasifibration

aha

what's a quasifibration?

that's the definition

oh I see

it's not very useful, but things like Quillen's Theorem B tell you that certain things are quasifibrations

well, cocartesian fibrations aren't quasifibrations in general - only in some cases, like when all the fibers are contractible

so maybe there's something more precise that can be said in this case

indeed, Quillen's Theorem B says that they are quasifibrations when the pullback/pushforward functors are weak equivalences

@SanathK.Devalapurkar @SaulGlasman did u guys figure this out? srry i was out of town

yeah I think we did

how do you guys bookmark some conversation? I see a "bookmark" option and I've used it today but it seems that what people bookmark is seen by everybody. That doesn't seem very practical. Maybe I should just copy-paste whatever I like somewhere

I was asked by an editor what keywords to put on a survey of (oo,1)-categories I wrote and realized I don't know what sorts of things people list as keywords. (I'm not sure I've ever noticed those keyword lists before, they appear as footnotes on the first page and start with "Keywords: "). Any suggestions? The paper in question is math.harvard.edu/~oantolin/papers/infinity-survey.pdf

What journal is this for? I didn't know people published expository papers at this level...!

As for keywords: the easiest thing to do is to find related papers and see what they used

Sadly, I don't think journals do publish stuff like that. This is for a conference proceedings, @ZhenLin

ah, still, a publication is a publication...

I had a paper I presented at a conference that I hoped to submit for the proceedings, but for some reason they didn't organise any

I thought the correct saying is a "a publication is still a publication, unless it is expository, in which case you are a sucker for writing it".

I could have had a publication a year ago if I submitted something to a conference proceedings when they asked for it, but I was too lazy... terrible decision

Exposition is a valuable service!

Valuable to others, possibly, not particularly to the author's career, right? I like expository writing and hope to do more of it in the future, but it does seem underappreciated.

If you write expository books it will probably count toward your teaching, somehow...

That's a good point, but books are a lot of work. :)

By the way, have you ever thought of publishing your notes?

Maybe someday, after I sort out my research...

(I got heavily criticised for not getting papers published properly in a grant review.)

Exposition is lovely, but my advice is: don't spend more than 20% of your time on it before tenure.

(Yes, I have a breakdown of how much time one should spend doing each thing.)

What's the rest of the breakdown, if I may ask?

That sounds sensible, Clark.

@Adeel Before tenure: 45% research/writing that you think you can probably complete in the next two months; 20% research/writing that is optimistic/long-range; 20% talks and written exposition for your own work or for the advancement of the way of thinking that you want to sell; 13% teaching and any mentoring that doesn't advance your own research; 2% service stuff such as referee reports, committee work, etc.

interesting, thanks!

Sorry ... I give lots of advice, because I got virtually none in grad school...

But, critically, I don't separate the research from the writing. The two are linked.