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?
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
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
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!
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.
@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...
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
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"
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.