12:30 AM
@SaalHardali i'm not sure that there's going to be a neat condition. you need the induced multiplication on the fiber I of R -> S to be trivial in a very strong sense, because R is going to be a "square-zero extension" of S. but homotopically this is data (chosen nullhomotopies of the multiplication I^I -> I and the bracket I^I^I[1] -> I and so on...) and using conditions to encapsulate the possibility of that type of data is often difficult.

7 hours later…
7:16 AM
@DenisNardin Nice, thanks!

2 hours later…
9:01 AM
@TylerLawson
Interesting. Maybe what I have in mind is a stronger notion that implies "small extension" but which is a property? Specifically, we can always construct a square of simplicial rings like so:

$$\begin{array}{ccc} R &\to& S\\\downarrow & &\downarrow\\ R^{\times_S \bullet + 1 } &\to& S \oplus TAQ(R \to R^{\times_S \bullet + 1 } \to S) \end{array}$$

Where I'm using the notation $TAQ$ from your chapter in Handbook of Homotopy and where the bottom map is the unit of the corresponding adjunction associated with $TAQ$. Then we can ask that (a) this is a pullback levelwise (b) the c
I arrived at my guesses from earlier by noting that the pullback can be checked in $R$-modules which by stability is the same as being a pushout which is equivalent to the map on the cofibers being an equivalence. Then you just verify that the cofiber of the left map is the (additive)-bar construction on the fiber of $R \to S$ (the cofiber of the right map it's obvious).
"... square of simplicial $R$-algebras"

9:20 AM
In his original notes on quasi-categories Joyal conjectures that an inner anodyne map of simplicial sets is equivalently a monomorphism which is a categorical equivalence and bijective on 0-simplices. In the preprint arxiv.org/abs/1810.05233v1 Danny Stevenson offers a proof of this claim, while Alexander Campbell provides a counterexample in arxiv.org/abs/1904.04965. Does anyone know anything about the current status of Joyal's conjecture?
I have only looked at both papers quite superficially; Campbell's counterexample appears to be quite simple, so it seems reasonable to suspect that there is a mistake somewhere in the more elaborate argument of Stevenson.

@AdrianClough IIRC There was a mistake in Danny Stevenson's proof

@DenisNardin Ok, thanks.

9:37 AM
In hindsight, I should have written an Acknowledgements section and thanked Danny for conversations on the subject of the paper (naturally I showed the counterexample to him before I wrote the paper), to prevent any such confusion.

It would be interesting to know whether Stevenson's arguments are strong enough to prove the following: A map of simplicial sets is flat (in the sense Lurie) iff it is an inner fibration and it pulls back inner anodyne maps to inner anodyne maps rather than categorical equivalences. This would obviously be true if Joyal's conjecture were true.
Exponentiable fibrations are are a joint generalisation of left final and right initial fibrations (see Ayala-Francis); the latter are precisely modelled by proper and smooth maps respectively, which themselves have an obvious joint generalisation: maps of simplicial sets which pull back inner anodyne maps to inner anodyne maps.

@AdrianClough I don't know whether that's true or not. You might be interested in the class of "absolute categorical equivalences" introduced in the Appendix to arXiv:1911.02631, which can be used to "correct" the characterisation of inner anodyne maps.

9:52 AM
These maps would then only differ from Lurie's flat morphisms by being required to be inner fibrations (which may thus be a superfluous condition). But this might all be a bit of a pipe dream.
@AlexanderCampbell Thanks!
@AlexanderCampbell I was looking for this appendix the other day, but couldn't remember what paper it is in (-:

12 hours later…
9:28 PM
Hi! Does anyone know if there's a partial/concrete answer or literature about the following question: consider doing unstable homotopy theory over a Hopf operad H in positive char p, so we may e.g. consider the category of (unstable) commutative algebras in the category of left H-modules.

It is easy to see that this category is controlled by an operad which I will call Com_H, so it makes sense to consider the Andre--Quillen cohomology of these commutative H-algebras, since we're in particular taking commutative algebras we need to work with simplicial methods (like Turner--Goerss).