@DenisNardin that's a great quote, thanks for sharing it

@HarryGindi Assuming Lurie's comparison of scaled simplicial sets with categories enriched in marked simplicial sets is correct (I've never looked at the proof) then you can combine that with the equivalence to infinity-categories enriched in infinity-categories (arxiv.org/abs/1312.3881), which are in turn equivalent to 2-fold Segal spaces (in the same paper), and thus to other models of (\infty,2)-categories (e.g. by appealing to Barwick--Schommer-Pries).

That's probably too indirect to give you any handle on the Gray tensor product though!

@Gasterbiter "Lurie will write it all up eventually" - As far as I am aware (which is not very far) there is no indication that Lurie is planning to write up foundations for (infty,2)-categories. I also think it's a bit disrespectful for (potential) users of the theory to reduce the community working on higher categories to just Lurie (even if his contributions so far are of course monumental).

@RuneHaugseng So something interesting: In the multitensor lifting papers, Batanin, Cisinski, and Weber proved something called the dropping and lifting theorems, which allow you to go back and forth between multitensors on globular sets, globular operads, and multitensors on strict omega-categories

I wonder if you could apply the strategy you used in the paper comparing n-fold segal spaces and theta_n spaces to the globular operad associated with strict omega-categories with the lax tensor product

Objects would be something like "categories weakly enriched in Theta-spaces with the lax tensor product" and you could try to recover it as an E_1 structure by desuspending

model-independent

Their series proves much more general things but you can use it to do what I just said

not sure how to prove that the associated 'monoidal product' would be E^1/hoassociative or biclosed, but I would have to read through your paper in more detail

@RuneHaugseng see math.harvard.edu/~lurie/papers/Ambidexterity.pdf remark 4.2.5

@AaronMazel-Gee I don't know, possibly. I was just pulling two random reasonable definitions of the (∞,1)-category of (∞,2)-categories to show that both being defined without model categories does not mean that two things are obviously equivalent. And there is a subtle but important difference between internal category objects and enriched categories (as in, they're not equivalent in general even though they are pretty close)

@ClarkBarwick obviously no one disagrees that theorems ought to have written proofs. where i feel youre crossing the line is when you openly bash a book which took the authors years of hard work to write and is an important contribution to the community. and yes i do consider it to be bashing, when you call someone’s book “appalling” multiple times.

whatever problems there may be in the culture of homotopy theory, i don’t think creating a culture where it’s ok to bash people’s papers in public chats like this, is the solution. anyway ill stop here too

I'm sorry, but I must, it seems, clarify: 'to appall', as I understand it, means 'to dismay to a great extent'. I was, indeed, profoundly dismayed by the book, as were others. I'm sorry if this word is misinterpretable.

I apologize if this is a profoundly naive question (or a profoundly difficult one, and hence naive), but what foundations need to be laid for Homotopy Theory to stand on solid ground? Put differently, what would go into a "homotopical EGA/SGA" that has not already been written or done yet?

@ReubenStern That depends on what you mean by "homotopical EGA/SGA"

@ReubenStern Model categories are rock solid, as are most models of infty,1 categories. Spectra are rock solid, G-spectra are partly translated to the infty,1 setting

(∞,1)-category theory is pretty much on solid ground for most of the things you might need to do. When you start to involve (∞,2)-categories, we are just starting to organize an amorphous blob of partial results in what could be a sensible theory. (I do not mean to disrespect people producing the aformentioned amorphous blob of results: they did a great job in a difficult area but there's still a lot to do)

I don't think that splitting between model categories and (∞,1)-categories makes any more sense. I think that pretty much any expert can with little effort translate results from one setting to the other

Enriched infty categories and infty,n cats have very little proven

Yoneda's lemma just proven this year

I would consider the equivariant and motivic theory on really solid ground. Yeah there's still stuff to do but you can use the basic theory without fear of stepping in an abyss

The major missing piece to even start doing really solid infty,n stuff is lax natural transformations and lax tensor products

In fact what holes are there in motivic homotopy theory come from the algebraic geometry side (cough - resolution of singularities - cough) not from the homotopy theory side

@DenisNardin not a fan of alterations, Denis ;)

@EldenElmanto Oh I am a great fan :). They don't solve everything unfortunately :P

without solid lax stuff, infty,n categories are crippled compared to strict n-categories

I know several people who would claim that equivariant homotopy for non-discrete groups is still lacking in good foundations

The compact Lie group case is pretty solid

@DenisNardin true - but distillations are lovely :)

It would be nice to have a BPQ theorem in that case, but that's not really a problem about foundations is it?

also, higher fibrations, I guess

In fact I'd go as far as saying that the infinite discrete group case is probably the most unsatisfying right now

@DenisNardin I don't know close to enough about this to have a useful opinion just quoting stuff i was told. btw didn't Hironaka upload a preprint of a proof of resolution of singularities for positive charactersitic almost a year ago? Whats up with this?

@SaalHardali It's... murky. It's not really my area of expertise though, so I shan't speculate

@SaalHardali we'll live with distillations for now.

I thought a consensus would have reached by now

I'm not senior enough to propose a program of research, but I would encourage people on the pure higher categories side to work on Grothendieck fibrations and lax anything. It's wide open..

@HarryGindi what do you mean that lax natural transformations are missing?

So either conjecture that the lax tensor product defined explicitly is a left-Quillen bifunctor that is homotopy-associative, or give a homotopy invariant version that preserves colimits argument by argument

or connect Verity's theory up with the rest of the corpus

the last idea is probably the easiest combinatorially

Regarding foundations I was told in this chat a while ago that the situation with enriched $\infty$-operads is still not perfect.

Bergner and Ozornova are working on it is all I know.

@SaalHardali Hold on, not perfect is not the same thing as saying that the foundations are broken. We have a reasonable definition of the theory. Yes we'd like to be able to work with it more easily but that's a long way from "the foundations are missing"

@DenisNardin Of course I agree. I even think you were the one who told me this.

@DenisNardin mathoverflow.net/questions/34673/… I should have thought about this. It seems my guess about the Cech complexes was right after all, so I must have mixed up some computation that led me to believe they weren't homotopy equivalent.

@PedroTamaroff The theorem above is only for quasi-coherent sheaves. OTOH asking for more is probably both unnecessary and way too general to hold

Silly question: does anyone know a proof of Atiyah duality that does not use that manifolds can be embedded in S^N?

I don't even know the statement of atiyah duality =)!

Let M be a compact manifold with boundary. The Spanier-Whitehead dual of $Σ^∞(M/\partial M)$ is the Thom spectrum $M^{-TM}$

(it should be thought as the spectrum-level version of Poincaré-Lefschetz duality)

@DenisNardin Sure, I am not worried about a more general setting. :)

@DenisNardin I have a weak memory of the duality chapters in May-Sigurdsson containing a pretty abstract setup in a bicategory of parameterized spectra, but maybe in setting it up one needs Spanier-Whitehead at some point. One benefit is they get a nice relative version of the statement.

But I never understood those in any serious detail so this is only a suggestion of somewhere to look; I do not want to misrepresent the contents.

@SaalHardali I was recently told by birational geometers that some experts have indeed looked at Hironaka's paper and found it unconvincing (as in, some of the proofs are problematic), though it does contain some nontrivial reductions.

@DenisNardin It seems unlikely to me such a proof already exists. (It also seems unlikely such a proof can be found, but I am much less confident about this.)

@DenisNardin @OmarAntolín-Camarena what if we tried to use the trick of writing M as a homotopy colimit of finite disjoint unions of disks embedded in M, to reduce to the case of a disk? After all, 'nonabelian poincare duality' ought to be related to Atiyah duality ;)

Is it true that if $2$-dim planar sections in some neighborhood of every point on a compact surface in $\mathbb{R}^n$ is a circle ($\mathbb{S}^1$) then the surface is spherical?

@DylanWilson Yeah, I was thinking that Poincaré duality does not use it at any place, so Atiyah duality shouldn't either

But then we'd need a version of Atiyah duality for open manifolds

well, just for disks right? which we have. and morally, I guess this must be behind the argument in 3.6.3 of arxiv.org/pdf/1409.2857.pdf

I think in order to run the argument about homotopy colimits you might need it for all (second countable?) open manifolds

I'm not sure I understand why... where are these other manifolds which aren't disjoint unions of disks appearing?

I'm a bit distracted right now, but I thought that to run the argument you need a natural transformation between functors that commute with homotopy colimits. But maybe you can just ignore that and prove by hand that the homotopy colimit of your functors is the right one