If I remember correctly, the suspension spectrum and infinite loop space adjunction has this property for model categories of highly structured spectra

Also, the Quillen equivalence between the Joyal model structure on simplicial sets and the Bergner model structure on simplicially enriched categories is oplax monoidal wrt the Cartesian product and the comparison map is a weak equivalence, but iirc the Bergner model structure is not technically a monoidal model category

@SaalHardali @DenisNardin I'm actually a bit anxious about switching the m and the n like that? For instance, a loop space is compatibly an E_1 algebra and an E_∞-coalgebra, but certainly not the reverse?

@JonathanBeardsley I think that was a typo, and that he meant to write Alg_P Coalg_Q = Coalg_Q Alg_P

Ah.

Yeah I think it's true

I see Dylan and you already talked about it quite a bit

Unconclusively, unfortunately

So I think for arbitrary P and Q you definitely need conditions on P and Q

Well, I guess, if you can do the BV product for ∞-props that will do it

For it to make any sense even

Oh I see

Yes

And apparently it exists (and is symmetric) for props in sets

I believe it

Is there work besides Hackney-Robertson-Yau on ∞-props?

See, I didn't even know about it :)

E.g. for quasicategories and Lurie style ∞-operads?

That's why I was hoping you had words of wisdom

Well, some.substantial translation would need to be done to use HRY in luries setup

Sorry typing on phone

No problem. I see they do have a BV product, so that's encouraging

Yeah I think at least for properads, maybe wheel free or something, they have a pretty nice category

It would be really nice, although maybe not worth the effort, to have some prop and properad tools that interacted with Lurie's work

I agree

I recall @ClarkBarwick having some thoughts on that once upon a time

I think he and Saul and I even had a slack channel devoted to it or something... But it never got off the ground

Maybe a good thesis project for some smart person is to figure out what the "indexing category" for bialgebras is, or something. How to BV product Fin_* with its opposite correctly

Or the bi-associative version if you prefer, with ∆ or something

Yeah, I can imagine this being a good thesis project. It's unlikely you end up with something undoable, but at the same time there are clearly hard details to be worked out

It does suffer a bit from the "you just proved the thing that everyone expected to be true" phenomenon

Yeah. Exactly. Although I think what I said, the question of the actual combinatorial description of bialgebra structure, would be fascinating to know

I don't think that's known?

Well, *higher bialgebra structure

I could try to ask Bruno Vallette. He seems to know combinatorial descriptions for everything

They all end up being horrifying graph categories though...

Haha. Yeah the issue is that it's sort of like taking the BV product of categories of operators. Or taking the category of operators of a BV product of an operad and a cooperad. Neither of which makes sense.

Yeah. And then all of the sudden Kontsevich is there and you have to shield your eyes.

I don't know if people in this room know about this, but the following is a pretty rad looking conference happening soon in the Pacific Northwest: pims.math.ca/scientific-event/190610-pwat

That looks super cool. I'm wondering if I can find a way to go there...

Oh and @SaalHardali in my paper "Thom objects are Cotorsors'" there are some really elementary manipulations with coalgebras and bialgebras in ∞-categories that might help stimulate your brain. It's probably got some mistakes in it, and I don't think your question is actually answered there, but it can help to just have some ground to stand on.

@DenisNardin I'll be there. It would be great to see you. And you could meet my daughter!

@DenisNardin @JonathanBeardsley @DylanWilson Thanks everyone!

@JonathanBeardsley Unfortunately my travel budget doesn't quite get as far as the Pacific Northwest...

Anyway, I've got to go. Nice talking to you!

@DenisNardin 😥

Has anyone carefully gone through Farjoun's proof that localization commutes with loop spaces? It seems as if it is incomplete unless we have an extra property, that the factorization of a loop map through the coaugmentation map (which is also a loop map) is a loop map.

I am unsure how to prove that extra part

@HarryGindi I'm pinning you just because i've seen you talk here about cisinski's book before. Is it a fair assessment that cisinski's book is roughly comparable to the first 4 chapters of HTT +- epsilon? Say I wantedd to learn from cisinski's book and then pass to HTT for topoi and presentable categories, starting from chapter 5+epsilon, would that be wise?

Its just I only just discovered it and I think its written terrifically.

@SaalHardali No, I think chapter 7 has some completely new results

and it doesn't really have any of chapter 3 of HTT

Cisinski said that his whole goal with the book (and the projects following it) is to develop the theory without using the coherent nerve and realization at all

so he proves unmarked straightening and unstraightening first as an almost completely formal consequence of building a 'moduli object with universal fibration' representing right fibrations

and then he proves in chapter 7 that the moduli object in question is equivalent to the ∞-categorical localization of the category of simplicial sets at the right anodynes

He's working with Hoang Kim Nguyen on extending this approach to the marked case

I guess the question is can I use what I learn in that book to read the parts in lurie about topoi and presentable categories without too much extra effort...

no

definitely not, I don't think

Aha :(

I mean, Lurie does everythign with Cartesian fibrations, and this is the language used in the literature, right?

since Cisinski's book doesn't cover Cartesian fibrations, you won't be able to just read his book and then go into reading the literature

If I can recommend, you should probably skip the proof of S/U on a first reading. It is quite complicated and it can feel a bit unmotivated without knowing the applications. Of course you should try to understand the statement very well. And also, marked simplicial sets are very important on their own, so don't skip them

I still recommend reading Cisinski's book before reading Lurie

The proof comprises the bulk of section 3.2 of HTT. Also, skip the construction of Kan extensions and read it from Jay Shah's "Parametrized homotopy theory: parametrized limits and colimits"

it's not that long, and he proves a lot of stuff that Lurie omits

I see. There's a very nice section on kan extensions in Cisinski's book looks like.

@HarryGindi So cisinski's book is not comparable to any combination of chapter's from HTT?

I'm not asking whether they are equivalent as I realize their proofs are wildly different.

For example yonedda embedding is different i imagine

Question is whether they are comparable regarding the abstract "invariant" theory

Aha, that's still helpful to know.

Anyway me and my friend hot to chapter 3 before discovering cisnski's book and now we think it will be much easier to read the rest of HTT after Cisnski's book.

Ch.1,2, Locally presentable infty cats, adjunctions, and cofinality I think?

Also, Kan extensions

It also covers a lot of the necessary material from e.g. Goerss-Jardine that is just assumed in Lurie

@HarryGindi Thanks! I'll think about it