@DenisNardin ah and one way to interpret this is precisely that Koszul duality will recover EVERY algebra because every algebra is augmented

In fact I suspect only semi-Cartesian is necessary...

Is the homotopy category of sheaves of S^1 spectra over Sm_S (smooth separated finite type schemes over S) with etale topology compactly generated by the schemes? If not true over general S, under what conditions is this true?

Something out of the blue:

Is Schur index theory secretly the equivariant Adams conjecture over the point?

ncatlab.org/nlab/show/Schur+index#EquivariantJHomomorphism

Here is what I mean with more details:

@DenisNardin do you have any idea what the monoidal envelope of $LMod^\otimes$ is?

I want to say something like $\Delta_+\times \Delta^1$, but I'm not sure...

@DenisNardin btw did you ever get around to putting together that "must read papers in homotopy theory" list?

Does anyone know if the following has been written down somewhere? I have two symmetric strict monoidal categories C,D (i.e. strictly associative, not strictly commutative. In my case they are enriched over chain complexes.) with the same objects X. I want to construct the universal symmetric strict monoidal category with objects X admitting symmetric monoidal functors from C and D, so a coproduct in a certain category.

I expect that one could write down explicitly what the morphism spaces of this category would be, but it should be a combinatorial nightmare.

@JonathanBeardsley Is think this is the category of "partitioned ordinals": objects are finite totally ordered sets equipped with an initial segment and morphisms are order-preserving maps sending the initial segment to the initial segment

@TomBachmann I never did, but OTOH no one sent me any suggestion... I'm still open to do it if there's interest but I realized it is sth a bit too time-consuming to just do it on my own

@DenisNardin Noone sent anything? I didn't see that coming... Ok nevermind then.

I have my sketches and notes but I'm not surprised I didn't get answers. I'd probably have to personally email people. Or at least send an email to Algtop-L

Hmm so here's a question I might post to MO.

Given an augmented algebra, maybe with some kind of "connectivity" condition, in a monoidal ∞-category, we should be able to take its bar construction and get a coaugmented coalgebra from which we can recover the algebra. This coalgebra should be encoded as a simplicial object satisfying a (non-Cartesian) Segal condition. And in certain cases this should give an equivalence of ∞-categories between augmented algebra and (certain) simplicial objects. Any ideas how to prove this?

@Grish this is true if you hypercomplete and assume that every smooth S-scheme is locally of finite étale cohomological dimension (and the compact generators are the schemes of finite étale cohomological dimension). This is true for example if S is of finite type over Spec(R), where R is a field of finite virtual étale cohomological dimension, or a strictly henselian noetherian local ring, or the integers.