@user40276 The localization functor is left-exact? Is this what you're asking?

@HarryGindi I'm not sure on whether I've got your point... I'm interested in knowing if there are any sufficient conditions to impose on a localisation in order to guarantee that the respective reflective subcategory will also be an \infty-topoi.

@user40276 Harry answered correctly. The condition is for the localization functor to preserve finite limits (i.e. L is left-exact)

@DenisNardin @HarryGindi Ah, ok! But that is kind of trivial. My bad, I should have asked the more specific thing that I had in mind. Suppose that I want to invert all the morphisms $X \times A \rightarrow X$ for a fixed $A$ like in the category of motivic spaces. Is there any reasonable condition on this localisation that would lead to a \infty-topoi. I know that this still somehow too general, though.

So you're asking what's the condition on the generating L-equivalences that makes the localization left-exact?

I don't know if there is a general answer for that

(motivic spaces are not an ∞-topos, btw)

@DenisNardin Is there an infinity topoi of unstable motives?

@TimCampion hmm, weird. thanks for setting me straight on that. i doubt it's a right bousfield localization though, because that'd require the fibrations to agree. i guess this is what @HarryGindi is saying, that the Quillen--Serre and Strom model structures are "not directly comparable"

@HarryGindi i agree that it's unfortunate that not everything in gaitsgory--rozenblyum is proved there. but i don't think it follows that it's "not a good source". of course, it depends what one means by "good". but note for instance that those unproved lemmas are certainly expected to be true; there's a big difference between this and if i wrote a proof of the riemann hypothesis and left out the hardest parts.

for whatever it's worth, nick has told me that he knows how to prove everything that's asserted there. i've encouraged him to add those proofs in, but in the end it's up to the authors themselves. of course, you could also take this as an opportunity to fill the gap yourself!

@RyanKeleti for whatever it's worth, i'm not proficient with any model category of spectra -- i never got past adams's blue book as far as point-set models go

@AaronMazel-Gee I'm going to be annoying and claim that this is even worse than if they stated them as "conjectures". You either have a written proof or you don't get to claim the results as true

@HarryGindi @DenisNardin there's also a completely model-invariant description of genuine G-spectra (for G a compact Lie group) in my joint paper "a naive approach to genuine G-spectra and cyclotomic spectra". this records them in terms of their geometric fixedpoints (as opposed to their genuine fixedpoints, which is the "spectral mackey functors" perspective). this builds on saul glasman's work (among others), which applies to finite groups.

@DenisNardin sure, i could agree with that. in any case, i don't think this makes the book objectively bad.

@SaalHardali yeah, i've heard that as a rule of thumb too -- i'd love to know a more precise statement though

@AaronMazel-Gee I just discovered this from common sense in the case of rational homotopy. The Koszul dual to the homology complex as a cocommutative coalgebra is the lie algebra model whose homology are the homotopy groups of the space. The universal enveloping of this lie algebra gives the homology of the loop space. In the other direction you can forget the cocommutative structure and take the cobar complex as an coassociative coalgebra to get the homology of the loop space.

@DenisNardin i believe @MarcHoyois asserted in a recent talk that every grouplike $A_\infty$-space is presentable as a compact Lie group. (am i getting that right?) of course, if you're only considering topological groups up to weak homotopy equivalence, then the notion of "closed subgroup" is not well-defined. are you thinking of things like the real line, or things like profinite groups, or anything in particular?

@AaronMazel-Gee I'm not considering them up to weak equivalence, that's the point :) Although I am a bit skeptical of the claim that every E_1-group can be presented as a compact Lie group: what about Z?

@SaalHardali gotcha. i've seen this asserted in the case of E_n-algebras as well, though i can't remember where. (and reall, what other operads are there, anyways?!)

And Z is exactly the kind of example I'm thinking about for the question

@DenisNardin haha okay yeah, i'm definitely misquoting the result

maybe it's required to be finite/compact as an $\infty$-groupoid

i've been convinced by a few people that the right place to start looking for the "right" theory (e.g. for noncompact Lie groups) is with manifolds with G-action

i can't remember the details, but i think the gist was just "what the heck do you ever really want to do with genuine G-spectra, at the end of the day"

Sure, I can get behind that. But it never seems to translate in actual concrete statements about what kind of transfers I should expect

yeah, i've never thought about the question itself -- just the question of how one might best address it

I agree with Denis and Harry. The GR book is pretty appalling. The issue is one I address in my notes on the future of homotopy theory. It is unhealthy for a subject to use one's high status as an excuse to go around claiming results as 'theorems' when you can't offer even a sketch of a proof of the fundamental work. It undermines the hard work of others to get the foundations on a good footing.

There were a lot of us who were trying to get the foundations of, say, the functoriality of the functors f^! on a good footing for many years. There was no question about how it would go once the 2-categorical stuff was in good shape. It never occurred to any of us that you could just say, 'the 2-category stuff is formal, we can't prove it, but that's not important', never cite any of the people who spoke to us about this stuff, and go on our merry way.

@AaronMazel-Gee I mean, I tried to prove these lemmas in greater generality for a project I was working on, and I wasted about 6 months and made no progress. Dimitri Ara also tried to find a pattern there, as has Cisinski, and I don't think we're all idiots.

it's not the identical setting, but it's related. A student of Dom Verity is giving the 2-dimensional case a shot again, but he didn't seem to have any new strategy when I spoke to him

The combinatorics seems to be so wild that it might be easier to try to transfer the monoidal structure across a Quillen equivalence. I did a calculation showing that it's not on-the-nose associative a while ago

(the conjectured equivalence is between complicial sets and Theta-spaces, because the lax tensor product is well-defined and homotopy-associative and homotopy-biclosed in the complicial case)

@DenisNardin There's a new paper by Anel, Biedermann, Finster, and Joyal that gives a condition for the maps generating the localization to give a lex one

Biedermann was just here at Regensburg last week and he gave a talk about it

^ Maybe you should also ping @user40276 :)

@user40276 There's a new paper by Anel, Biedermann, Finster, and Joyal where they define something called a 'modality', which is a certain kind of (∞-categorical) orthogonal factorization system where the left class is closed by pullbacks

they then give conditions for the left class to be a 'lex modality', which generates a lex localization

they also gave a neat application, recovering the unstable goodwillie tower from this stuff.

@ClarkBarwick right, i'm definitely on board with the fact that the book ought to contain proofs of those facts. perhaps a closer-to-home assertion is just that i've personally learned a lot from it, and to say it's "bad" seems not so different from saying that "we'd all be better off if it didn't exist". as for our usage of the $(\infty,2)$ material there, i'm pretty sure that it didn't rely on any of the unproved assertions; i wouldn'tve felt very comfortable with that, either.

@AaronMazel-Gee Is there any ∞,2 stuff in there that doesn't rely on the lax tensor product? The entire discussion of 2-straightening and etc. relies foundationally on it

@HarryGindi the issue is the monoidalness, right?

no

homotopy invariance

it's not known to be either a left quillen bifunctor (which is what you need for homotopy invariance) or associative (or homotopy associative)

@DenisNardin ah, so actually i have recently come across some form of an answer to this in some stuff i've been thinking about. namely, just as THH is a genuine-proper S^1-spectrum, 2-fold iterated THH is something like a "genuine-proper spectrum with respect to the group of framed automorphisms of the torus". the precise definition is through the "stratifications" picture, of course

@HarryGindi everything in there is by definition homotopy invariant, so i'm not sure what you're referring to

then the existence of adjoints

you can define it to be homotopy-invariant easily enough but then it's not clear how to show that it is biclosed

i'll have to go back and look, but i thought it was defined through a left kan extension?

anyways, thanks for pointing that out -- i will check back with nick about the status of the "proofs that will be supplied elsewhere"

the part where it says that it preserves colimits argument by argument says "we do not provide a proof and were unable to find a reference" in the footnotes

gotcha. where's that?

page 18 of the first appendix

(or i can find it if you don't have it open)

okay, thanks

math.harvard.edu/~gaitsgde/GL/Basics.pdf

if all you need is the lax morphism objects (and not 2-straightening), that paper of Johnson-Freyd and Scheimbauer does construct them with a great deal of effort

@ClarkBarwick to be clear, i do agree that it's unhealthy for the field to have assertions with this status -- "proof by fiat" or something

@AaronMazel-Gee If you're close with Nick or know him well, you should impress upon him that there are a bunch of people who would like to see the proofs of these statements and not for the purposes of mere curiosity

I'd personally like to see them not just for comfort but because I couldn't figure it out after a lot of effort

I said it was appalling, not 'bad'. I might also say embarrassing; I'd love not to have to talk about this. However, it's a prime example of how no make things much harder for the rest of the community by clipping off the easy-to-swallow and broadly-appealing assertions for oneself while leaving seriously hard and less popular 'engineering' work to others. I consider that unacceptable.

At the same time, it makes the work of people like Grothendieck or Deligne all the more worthy of celebration. The culture of celebration of the tough foundational work of algebraic geometry is very inspiring to me, and I'd like to see it similarly highly valued in homotopy theory.

@ClarkBarwick By the way, Dimitri Ara suggested in his paper on Quillen theorems Bn for strict n-categories that one of your papers with Dan Kan gave a construction of lax commas for weak n-categories. I couldn't really find what he meant by this

but interestingly enough, that should give you just enough structure to attempt straightening/unstraightening

Do you know what he was talking about?

by weak n, i mean infty, n

I don't know off the top of my head, but that may be because I'm insecure in my understanding of what one means by lax commas. Do please send me an email with more detail if you can, and we can see whether there's anything in my papers with Dan that might help.

alright, cool! Thanks!

Consider schemes $X',X''$ with open covers $\mathcal U',\mathcal U''$, and sheaves $\mathcal F',\mathcal F''$ of modules, respecively. In the product $X= X'\times X''$ we have a cover $\mathcal U =\mathcal U'\times \mathcal U''$ and a sheaf $\mathcal F = \pi^*_{X'}\mathcal F'\otimes \pi^*_{X''}\mathcal F''$.

It seems to me that in this case the Cech complex $C^*(X,\mathcal F)$ is just the total tensor product of the Cech complexes $C^*(X',\mathcal F')$ and $C^*(X'',\mathcal F'')$. Is there any generalization of this to pullbacks of sheaves, in the lines of the Eilnberg--Moore theorem for pullbacks in spaces? I would suppose this would work for constant sheaves, and that the theory would be much more complicated for arbitrary sheaves.

@HarryGindi @TylerLawson I've asked a question on the main site about the issue of cofibrant replacement of the unit of an algebra that we discussed three days ago: mathoverflow.net/questions/315699/…

Cool! upvoted!

@HarryGindi haha yes, this conversation has reminded me to do that -- in fact, i'm among those people too

@ClarkBarwick oh, sorry for the confusion -- i was referring to the original comment (harry's). anyways, i totally appreciate what you're saying!

does anyone know where i can find a (stable) adams chart for the cone on eta?

Hi! Apologies if my question above is slightly off topic here, but I couldn't think of another active chatroom in MO to ask. Any pointers or comments are helpful. :)

@PedroTamaroff Nah, don't apologize. We're fine with that kind of questions here

Just, sometimes no one around knows anything relevant. Disappointing, but happens (haven't looked at the message yet)

@PedroTamaroff when you say "generalization of this to pullbacks of sheaves", what are you referring to? (it sounds like your sheaf on X is already pulled back from X' and X''?)

@AaronMazel-Gee Here I'm taking the pullback of X' ---> pt <---- X''. So I mean a general pullback diagram, perhaps with some hypothesis on one of the arrows?

@AaronMazel-Gee thanks for your answer on the topic. However, I don't really follow: I'm not asking that a cofibrant replacement $QX\to X$ factors through the point

Recall E-M tells you that H(E) is Tor_C*(B)(C*(X),C*(Y)) if the diagram is X > B < Y and E is the pullback.

@PedroTamaroff Yes, in your setting what are the sheaves?

I think that in order to generalize the E-M theorem you'll need your sheaves to have some kind of ring structure

@DenisNardin In the setting above? I take a sheaf on X' and one in X'', pull them back to X by the projections and take their product.

No, in your "generalized" setting

I think you'll need at the very least a sheaf of rings on B and structures of modules over the pullback for the sheaves on X and Y even for the statement to make sense

Ah, I would assume that in the generalized setting one would have to do the same, but take the tensor product over the structure sheaf O_B?

What's the structure sheaf? :)

Oh you want to work with sheaves of modules over schemes

Yes, exactly.

So, in derived-speak, you want to know when is the map $f_*F_X⊗_{O_B}g_*F_Y→π_*(π_X^*F_X⊗_{O_{X×_BY}} π_Y^*F_Y)$ an equivalence in D(B)

So the map is clearly an equivalence when $F_X=O_X$ or $F_Y=O_Y$

The proof of E-M usually works by filtering $F_X$ and $F_Y$ by objects of that form (or similar)

Yes. You produce a quasi-free resolution and show you get differential Tor. But then in sheaves we don't have that. :/

I would expect however that the left hand side of your equation perhaps just needs to be replaced by something of that flavour, though?

is it true for flat resolutions?

Not sure.

No wait. Sorry I said a stupid thing. The map is not obviously an equivalence when one of the sheaves is the structure sheaf

No worries. I am not really a user of schemes, I'm actually trying to think in terms of the Cech complexes. It seems to me that C(X,F) is bigger but nevertheless quasi-isomorphic to C(X',F') x C(X'', F'') in the case of a product.

And to be 100% transparent I was looking at the case of line bundles in P^1 and their pullbacks to P^1 x P^1 as a computational exercise. But this seems to work in general.

Ahh... the map I wrote is the wrong one sorry

The one you want is $\Gamma(F_X)⊗_{\Gamma(O_B)}\Gamma(F_Y)→\Gamma(π_X^*F_X⊗_{O_{X×_BY}} π_Y^*F_Y)$

Pedro, is that a Cartesian product or tensor product?

Which one?

Oh, the last one I wrote is the (total) tensor product of Cech complexes. But now I think that may not be right.

(Anyways, the claim about the Cech complexes is false!)

@ClarkBarwick i disagree with the Gaitsgory-Rozenblyum bashing. i appreciate what youre saying that the homotopy theory community is unhappy with them, but they did a huge service to the algebraic geometry and representation theory communities. there are a lot of people very happy about their book. keep in mind that the audience of the book is not homotopy theorists

@Gasterbiter At least in my opinion claiming that stuff is true while no written proof exists is fundamentally wrong. I understand that people might not care about such categorical technicalities we are talking about, but that's where often mistakes hide. I don't know how to trust any result in that book as a consequence. That is not a good state of affairs. They should have written in the book that all results are conditional to this and that conjecture

(or, even better, actually write down the proofs of the nontrivial lemmas)

I could write hundreds of "plausible" statements about higher categories that turn out to be utterly false. While I'm not saying that this is the case for any of the results in GR (and as a matter of fact, I believe them to be essentially correct), I cannot know it. And that's a terrifying way of building a subject

And the worst part is that if someone finally puts in the work and manage to prove the assertions, he will be doing it pro amore Dei: he'll receive no credit for it since "it was all already in GR" (while it wasn't!)

@DenisNardin What are the conjectures that are being used, by the way?

I don't know about all of them, but Harry pointed at the note at page 18 of math.harvard.edu/~gaitsgde/GL/Basics.pdf

I mean, they write "We do not prove it, and we were not able to find a reference.". But they don't even say whether they have a proof or not!

@PedroTamaroff there are dozens of statements that rely on the missing stuff

Right. I see lots of footnotes there saying they don't have a proof.

I don't know if they care, but my preprint proves the Yoneda lemma without straightening using model categories. there's also a paper by Hinich that proves it using Gepner-Haugseng enrichment

if their only goal is to have Yoneda's lemma without lax constructions, there are now two preprints claiming to prove it in different levels of generality

and different models

but all of this lax stuff is seriously difficult

also higher Cartesian fibrations

I encourage everyone to read this email by Mumford about the Italian school of algebraic geometry.

Lyle is definitely wrong

A very good mathematician that I know published a completely erroneous paper based on a sequence of completely erroneous assertions in a paper by a less careful mathematician

Do "we" believe that the list of unproved assertions in GR.A.1.0.4.2 is complete?

@HarryGindi @MarcHoyois ^

It is complete in the sense that those specific assertions are not proven

it is incomplete in the sense that all of the proofs following those proofs and definitions depend on them

Well, actually, I have no idea if it's complete. I'm prepared to believe it though

@DenisNardin i dont know about in homotopy theory but at least in AG, you might be surprised to know that "claiming that stuff is true while no written proof exists," is something that happens in a very significant percentage of papers. remember that mathematics is a social activity, you have to convince humans of the assertions in your papers, not computers.

imagine if someone proves geometric Langlands building in particular on GR's book, and at that point still nobody has written proofs of the unwritten assertions. i would imagine that the community will certainly accept the proof irregardless. sure people like you and @ClarkBarwick in the homotopy theory community may object, but I highly doubt the experts in geometric representation theory would find issue with the claims about infinity-2 categories

And that's exactly what I'm complaining about. Because you can get honest to God mistakes that way.

Now, if I were to stretch the analogy above I'd say that we are still in the "Enriques" phase of the problem: good mathematicians (surely better mathematicians than me) skipping annoyingly technical details. But that's how you end up with people proving crap twenty years later

my question is if homotopy theorists feel this way, then why doesn't someone write down a proof? Rozenblyum cannot be the only one capable of proving this. Lurie will write it all up eventually but it might be good for someone else to step up :)

Because people have tried and failed

I listed three people who have attempted with various degrees of intensity to give a proof

that surprises me. asking around, i found various people claiming that they had proofs of at least some of those assertiosn (some even planning to write them up). i cant give names here obviously

I eagerly await the proofs!

I'm not asking you to. But I've written proofs of similarly technical statements and I'm afraid of taking people at their word in this

@Gasterbiter an easier combinatorial proof about the extended B-V tensor product of dendroidal sets was wrong

the statement was wrong and the proof was wrong

and it took like 5 or 6 years before it was finally fixed completely

im not saying the proof is easy, i just dont think its as unprovable as various people make it sound here, at least thats my impression based on talking to some of my more homotopy theoretic colleagues

and if it has the same formal properties

in the worst case the statements are wrong, we throw out the GR book and every paper that cites it, and move on with math. thats fine to me IMO

@HarryGindi if its true in a model, then its true in their model independent setting. maybe the question is whether your model is actually modeling the right thing then.

I don't think they are "unprovable". But minimizing this stuff as "just (∞,2)-categorical combinatorics" is certainly underestimating the amount of work you need to do a great deal

@Gasterbiter "model independent" is kind of a joke in this case

I know that people don't like when I say that these issues are actual mathematical issues of substance that you need to work out. But they are.

@HarryGindi you lost me then. it sounds kinda like youre saying that you can prove something in the joyal model structure and youre wondering if its true in the infinity-category of infinity-categories. but maybe im not knowledgable enough to understand what you mean here

something is proved combinatorially in an explicit model for (∞,∞)-categories that is not known to be equivalent to any of the other models for ∞,∞-categories

also, even if you did have a Quillen equivalence, you don't necessarily have any idea what kind of structure can be transported, that needs to be proven as well

Aren't GR building (oo,2)-category theory in a model independent fashion out (oo,1)-category theory? So any model which cannot prove equivalence with GR is kind of flawed?

@HarryGindi sure, ok. well if people believe that this is in fact an explicit model for (oo,oo)-categories, that seems to be a very strong evidence that the claims are at least true

Sorry I might be off about this, it's been a while since I thought about these things.

@TomBachmann yeah that was my impression as well

Has anyone proven it's equivalent to Gepner-Haugseng enrichment, which is also model-independent?

yeah i think thats in one of Rune Haugseng papers

actually not sure, what im finding now is arxiv.org/abs/1604.08480 but this is different

yeah, that's totally different

that's a really nice paper though

What does "model independent" means in this context?

to my mind, 'model independent' should mean something like 'characterized by a (homotopy) universal property'

but it seems like oftentimes it's defined to mean "without reference to an explicit model category structure'

I mean, they define a certain (∞,1)-category that they call the (∞,1)-category of (∞,2)-categories but that does not prove that this coincides with all the reasonable (∞,1)-categories of (∞,2)-categories we could cook up

totally agree

@DenisNardin Can you point to roughly where this definition is in their book? Just interested.

I can look it up by myself in any case just hoping to save some time

@SaalHardali Appendix 1

In particular definition A.1.2.1

Thanks!

In my mind "model independently" more or less means "expressed in a (not explicitly written down) synthetic theory of (oo,1)-categories"

Well sure, but both "higher complete Segal spaces" and "categories enriched in (∞,1)-categories" fit the bill, and there's no real reason for them to be the same thing

(note: I totally believe them to be the same thing)

there are a few proofs now that they are the same

Right. I agree one should prove they are the same, and that this is an expected (but as far as I can tell non-trivial) theorem.

Bergner-Rezk proved this by a zig-zag, and my preprint proves it more directly

On a more cheerful note, D.C. Cisinski's answer here is great and everyone interested in motivic homotopy theory/algebraic K-theory should read it :)

as usual =)

I see I've been 'pinged' about this by an anonymous user. The extent to which various people are happy or unhappy is irrelevant to what I've said. My assertion is that the only things that should be called 'theorems' ought to have complete proofs. This isn't a community-dependent phenomenon, and this isn't an academic matter. Results about higher categories are nontrivial. Treating them dismissively, as if they were mere technical footnotes – as GR does and as you are doing – is unacceptable.

This isn't bashing, and I can't imagine why anyone would consider it controversial. As far as I know, it's the normal standard applied to mathematics, in all branches. For my part, I suspect that the unproved claims are correct. However, the proofs appear to be difficult. Now, that last sentence is a judgment call, and you're more that welcome to doubt my judgment. But if the proofs are NOT difficult, then it follows that it will be easy to prove me wrong. And I'll be ecstatic.

@DenisNardin does this follow from barwick--schommer-pries? i believe rune also proved that "category-objects in a (nice) cartesian symmetric monoidal \infty-category X" are equivalent to the gepner--haugseng model for X-enriched \infty-categories (in fact, this is true before imposing a completeness condition (in the sense of complete segal spaces))

(If anyone wants to continue a conversation about my opinions about this, you're welcome to email me. But let's stop talking here. I've said what I want to say about this publicly.) I seem to recall that Chris Schommer-Pries and/or one of his impressive array of students may have proved that the Verity complicial model satisfies our axioms for n=2. If I recall correctly, it's relatively easy to show that the GR theory does. So if you combine that with Harry's comment about the Verity model ...