@TylerLawson Absolutely, and even the same person can want many different things, depending on the problem at hand. The point is that some of the possible answers do not behave as you might expect/hope.
@EricPeterson no, I admit I was lazy and bothered you first, I thought I recalled you having an interpretation of the E-(co)homology of symmetric powers.
the goodwillie tower material was why I asked, related to this giving a factorization S^m -> Sym^p(S^m) -> ... -> Sym^{p^n}(S^m) ~= K(Z_p, m) in K(n)-local homotopy theory
e.g. I was wondering whether some of these maps were zero in E-cohomology
it's possible we've talked about this before, i can imagine it having come up in the same conversation as free E_infty resolutions of L_K(n) MU. but, if i once knew something then i can't recall it now
the long composite is null in E-cohomology once m exceeds 2. that's all i've got
I'm not 100% sure of what you want. You mean if for every diagram shape there exists a colored operad encoding the notion of "being an homotopy-coherent diagram of that shape"?
I think such a thing exists, but why do you need that operad?
philosophically speaking, why is it that all of the models for \infty-categories are the fibrant objects of a model structure? is it just because the models i'm thinking of have that all objects are cofibrant? or actually i guess relative categories don't have some easily describable distinguished class of "\infty-categories" among them..
still, can anyone think of a better reason than "all objects are cofibrant" why all of complete segal spaces, quasicategories, and kan-enriched categories appear as fibrant objects?
yes, i'm not sure. weren't we just recently discussing a preprint by lennart meier in which he shows that model categories are fibrant? maybe clark will correct me or have some useful insight, but i don't think there's a concrete description known. on the other hand, i sort of wouldn't consider a relative category as a 'presentation' of an \infty-category, in that it's not readily manipulated as such in the way that those other models are
so consider the question of localizing a relative \infty-category (C,W) to obtain (what i've been denoting by) an \infty-category $C [[W^{-1}]]$ where we've freely inverted the maps in W. every construction that i can think of (excluding relative categories for the reasons described above) involves first making some construction that leaves the world of \infty-categories and then fibrantly replacing.
yes, i guess come to think of it is does feel like it comes down to the fundamental handedness with which humans have approached this all (studying e.g. Sets instead of Sets^{op})
A model category of co-∞-categories could be constructed by taking a right Bousfield localization of a minimal model structure on a category of functors into co-sets.
But the choice of presentable has a chirality. We could have contemplated co-presentable infty-categories instead. There just aren't as many examples of interest.
mm, i'm thinking for instance of some argument about why projective (e.g. free) resolutions are so much easier for humans to understand than injective ones
@AaronMazel-Gee Yeah, so a concrete place where this comes up is the story of operads. Actual operads (in spaces, say) have a model structure, but the cofibrant objects are really hard to find in nature. So the model category of operads was hard to use (e.g., for the statement that E_n otimes E_m is E_{m+n}). The new, weak versions of operads (all built via Cisinski model structures) due variously to me, Moerdijk-Weiss, Lurie, etc. solve this issue handily.
@Adeel Sure. For a brief time, MO was really a place where you could ask a proper research question and get a technical response, or at least a literature reference. You can still do that, a bit. But now it's increasingly a place for pointless arguments between arrogant people about what's important and what's not. To borrow a phrase I learned from @Saul
MO generates more heat than light.
(The arrogant ones usually remain anonymous, I notice.)
For a while I wondered whether it was the points system that was to blame. But I think that it's more that people are using MO more for commentary than for actual work. I prefer to work than to read the views on "whether model categories are still relevant" from some guy on the internet.
But back to the question at hand: Sets are not self-dual.
@ClarkBarwick so, cofibrancy of operads is related to free actions of groups (if i'm intuiting correctly), which often require 'big' spaces to obtain. are you asserting that this behaves somehow like an injective resolution?
If I have a functor F:C--> D such that C is cofibered in groupoids over D, we can for various choices of fibers try to build functors between the fibers, which are groupoids. So say that we have d,d'\in D. According to HTT one should for a morphism f:d--> d' choose a morphism g : c--> c' covering f and set f_!(c)=c'. How do we define the functor on morphisms?
(in reference to me repeating somebody's assertion that projective resolutions are easier to think about than injective...but maybe "easier to think about" is not always the same as "easier to manipulate")
i thought the classical terminology came from the fact that a stack is supposed to be a special sort of category living over our base that's "fibered in groupoids", and these come with pullback functors...hence are right fibrations, i think. and then "cofibered in groupoids" just came to mean the dual thing
@ClarkBarwick this is an interesting point of view. I have had a lot of experiences on here where people seem more interested in quibbling over minor points in language than actually answering a question.
Or just showing off their knowledge about some tangentially connected but not really relevant topic.
I still get quite a bit from it, but it's sort of more reduced to the point of getting useful references or finding out whether or not big-shots think the answer to my question is yes or no
(but not actually how to prove it)
@ClarkBarwick well, mostly on MO. but also in this room.
there's a lot of really lovely discussion that happens in this room.
and there is of course something to be said for people encouraging me to stop being so goddamn imprecise. =P
But in general, yeah, it seems like it's the rare mathematician who wants to determine how what you're saying is TRUE rather than the multitudinous ways in which it is false.
Well one thing to say is that many of us seem to think about things in vastly different ways. Especially those of us that are not sort of, I don't know, in the same flow as others, or that get most of our ideas from weird sources.
@user101036 Complete Segal spaces are actually handy for constructing some infty-categories. The nice thing about them is that you specify a complete Segal space by identifying the moduli space of n composable arrows for each n.
And so that can lead to rather severe language barriers.
Like the other day when Saul said "intertwine" and I had no idea what that meant/.
That's not at all about arrogance, it's just about the fact that Saul and I haven't discussed that particular subject before.
So I think, in the defense of the people that I'm sort of indicting, there are times when one simply cannot understand what another person is trying to say.
@user101036 to piggyback on clark's answer, for a specific example, lurie constructs the bordism $(\infty,n)$-categories by producing it as a segal object (and then applying the "completion" functor)
@user101036 A good example is the category of spans in a category with pullbacks. It's very easy to see that it's a complete Segal space, and then you can use Joyal-Tierney to get a quasicategory. Proving directly that the thing you get is a quasicategory is doable, but less easy.
i also happen to use them in my own work, more precisely the notion of a CSS 'internal' to the theory of \infty-categories (by which i'll mean quasicategories, if i need to be precise): a full subcategory (a left localization, in fact) of the \infty-category of simplicial spaces
i learned this perspective in lurie's "goodwillie calculus" paper, but i'm pretty sure most of the ideas come from clark's thesis
@ClarkBarwick haha yeah agreed. I don't know. I think, like in every part of life, there are nice people who want to work together towards a better joint understanding, and there are not-so-nice people who want to be right as often as possible.
from a different perspective, I think that complete Segal spaces (at least up to Reedy weak equivalence) are important because they are a definition of (∞, 1)-categories directly inside homotopy theory
@Adeel sure, to be precise: any model category has an underlying quasicategory, and that of rezk's CSS model structure yields a quasicategory that's equivalent to the one coming from the Joyal model structure [ignoring set-theoretic issues, i'm sure]
@ZhenLin Right on! That's my favorite thing about CSSes and n-fold CSSes: all of the data you have to specify is a homotopy invariant of the object of study. That's a good thing.
@AaronMazel-Gee, right, I always say infinity-category "generated by" a model structure when people usually say infinity-category "underlying" a model structure. I always find the latter strange, I guess I must be thinking about things backwards...
I said "Quasicategories are really neat. It's wonderful that you and Rainer Vogt came up with them." And he said "Well, all we did was look at what was there and write it down."
Oh I know. I've mentioned that one to him as well, haha. I'm pretty sure that he again responded by implying he was doing nothing more than writing down just what was there.
It is funny, sometimes algebraic topologists come to speak at JHU and they meet him, but they don't realize that he's that Boardman until much later.
Speaking of which, does anyone know a good proof of the following fact: Given a morphism of connective ring spectra $\phi:A\to B$ such that $\pi_0$ is an iso and $\pi_1$ is onto, the fiber of the canonical inclusion $A\to Tot^n(B/A^\bullet)$ is $hofib(\phi)^{\wedge_An}$?
Carlsson proves this in a really really long way. And Lurie states it.
(where $B/A^\bullet$ is the so-called Amitsur complex, or the sort of formal dual to the Cech nerve)
@Adeel i like the word "underlying" because it implicitly asserts that the \infty-category is the true object of study
i may be missing something easy, but does anyone see why Ex^\infty commutes with filtered colimits? rezk asserts it here: mathoverflow.net/questions/12746/…
@JonBeardsley the version of the Dold-Kan correspondence for stable infinity-categories in HA should transport the cosimplicial object to its Tot-tower. I think that you can identify the layers as total homotopy fibers of cubical diagrams; I can't remember if Lurie is explicit about that or not. that might get a little more towards the core of the calculation?
Yeah perhaps, thinking about applying that Dold-Kan correspondence to the diagram of cosimplicial spectra defining the fiber. Perhaps.
(I mean, taking the relevant cospan in Fun(\Delta,Spectra) and then applying the correspondence to it and looking at the layers)
The really important thing for me is that Carlsson uses commutative ring spectra the whole way through, and I can't find a place that this is really relevant. So I started rewriting the proof for just associative (A_\infty) rings, and just felt like there must be a better way to do it.
you may not even need to use a diagram of cosimplicial spectra; the n'th layer in the Dold-Kan should be something like the "kernel of all the codegeneracies" in degree n and that may spit out the answer you want
in the thing on the other end of the Dold-Kan correspondence, I mean.
@JonBeardsley yeah, I think this is roughly what he's doing around Remark 1.2.4.3; he's working simplicially, but he's identifying the layers in the geometric-realization filtration with a cofiber of a map from a latching object. if you switch to something cosimplicial, it's identifying the fibers in the Tot-tower with fibers of a map to a matching object
so if i : I -> J is a full inclusion and F : I -> C is a functor to a category with colimits, then the left Kan extension i_! F is given by the formula
I mean... on one hand, those make English writing more precise. On the other hand, anyone who understands what that symbol means probably already knows how to say the word.
That's nice though. I hope I get a chance to see what students are like at more elite academic universities. My ideal world is just a place where everyone is invested in learning a lot and becoming better people. Unfortunately, the longer I've spent in academia, the more I've come to realize it's not quite the utopia I thought it would be.
I pretty much want to live in a Herman Hesse book, if that means anything to you.
I haven't really spent much time in the corporate world, so I guess I wouldn't know. I've spent brief periods involved in essentially corporatized pedagogy, and that's been miserable, but not because of the people involved.
Yeah, that's true. I like that, haha. I'd say a lot of the corporate, business oriented world is also moving in that direction however (at least, the upper echelons of creative people, e.g. programmers, engineers). That is, give people interesting jobs and you won't have to force them to be there at 8am every morning.
Here's a question to which I should know the answer: suppose I have a representation over C (say), and I take its associated Mackey functor in C-vector spaces. To what would the extra structure of a Tambara functor correspond?
I was just trying to apply the principle we discussed above ... that one should try to see ways in which a person is correct but unclear, instead of assuming they're a clear-speaking idiot.
We should write one giant, open-source computer program that lives on the cloud that starts out with just things like integers, and arithmetic and allows people to define new objects. Kind of like an interactive version of the nLab.
And then you can try to make statements, and it won't tell you if they're necessarily true or whatever, but it can tell if you if there are any obvious contradictions.
What I mean is that Coq makes "obvious" steps in a proof. So you can give it the negation of X, ask it to deduce that using trivial steps, and if it can, you know that X is triviall false.
Hm, but THH satisfies descent. That's weird, since THH(S)=S, so is the descent sseq going from THH(MU)=MU^SU_+ to S the same one as the descent sseq going from MU to S?
@ClarkBarwick oh okay. i guess somehow i was thinking that you use a TAQ etaleness, and in nice cases at least we can build THH from TAQ, so I was wondering TAQ-etale leads to THH-etale thru HKR or something
if one wanted to read Rognes's work about algebraic K-theory of ring spectra, with a minimal background in "classical" homotopy theory, what would be some prerequisites? could i jump into his Galois theory book and then into the K-theory, or do I need some more topological prerequisites?
i just realized i don't understand what the smash product of spaces looks like as an operation on, say, pointed infinity-groupoids. in particular let's think of S^n as the free infinity-groupoid on an n-morphism and S^m as the free infinity-groupoid on an m-morphism. how should i think about their smash product in such a way that it's obvious that it's the free infinity-groupoid on an (n+m)-morphism?
i think this is the thing i'm actually confused about in this question:
To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of manifolds on its homotopy groups. By Pontryagin-Thom, $M$ is the sphere spectrum $S$, which is even the...
@Adeel It depends on what you want to read. What John does is largely very computational, so you'll have to develop some familiarity with v_n-local homotopy calculations. Bockstein spectral sequences and the like play a big role.
So if X is an ∞-groupoid with a vertex x, specifying a pointed map S^n ---> X is specifying an endomorphism of id_id_id_..._id_x. (I'll just call this id^{n-1}_x for clarity.) The pointed ∞-groupoid Map(S^n,X) is indeed the space End(id^{n-1}_x), pointed at id^n_x. Now homotopy classes of maps from S^m /\ S^n ---> X are in bijection with homotopy classes of maps S^m ---> End(id^n_x).
These, in turn, are endomorphisms of id^{m+n-1}_x, or equivalently, homotopy classes of maps S^{m+n} ---> X.
I can't edit, but the End(id^n_x) at the end of the first comment should be an End(id^{n-1}_x).
@Clark: ooh. right, of course, this is just the universal property of the smash product. got it. thanks!
alright, then re: my bordism categories -> ring spectra question, the other thing i don't understand is what categorical structure a symmetric monoidal infinity-groupoid, to be thought of as a connective spectrum, needs to have in order for the corresponding spectrum to have a ring spectrum structure
so i guess i don't understand the smash product of spectra as an operation on symmetric monoidal infinity-groupoids, or something
let me take an example that already confuses me, which is finite-dimensional real vector spaces Vect. as a symmetric monoidal (topological) groupoid under direct sum this presents ko. so far so good
now ko is also an E-infinity ring space, and the multiplication should come from the tensor product
but naively the tensor product as an operation Vect x Vect -> Vect should give me something like a multiplication pi_n(Vect) x pi_n(Vect) -> pi_n(Vect), which is not what I want
what i want is an operation Vect smash Vect -> Vect which will give me an operation pi_n(Vect) x pi_m(Vect) -> pi_{n+m}(Vect) (here i'm using Vect for both the category and the infinite loop space)
so here I guess my conceptual confusion is that I understand what Vect x Vect is as a categorical object, but I don't think I understand what Vect smash Vect is as a categorical object
yeah, i was worried about something like that. the analogous issue already exists for a ring R where the multiplication R x R -> R isn't an abelian group homomorphism
