@Gasterbiter thanks that's very helpful!

@DenisNardin thanks! I've been thinking about it some more... I think I maybe have some moderately sensible things to say, but I felt like I'd already sent you enough e-mails

@skd hm yeah, this "propagating module" and "universal descent morphism" stuff is very interesting...

And yes, as @Gasterbiter says, it may be possible to re-do d 5.6.6, but that's rather too strong. that's showing that the map is an effective descent morphism (if I understand correctly) but in many cases we have effective universal descent morphisms which are not effective (for the module stack at least), again if I'm understanding correctly.

For instance $S\to MU$ or $S\to X(n)$.

Hm actually, I'm not sure I'm understanding what a "universal descent morphism" is...

Yeah I'm trying to piece things together here... and it looks like he's saying that if the unit map $u:A\to B$ induces an equivalence between $A$ and the limit of the Amitsur complex then... $LMod_A$ is equivalent to the category of $B$-modules equipped with descent data? I.e. this map is an effective descent map?

But how is that possible? For instance it seems to me that $S$ is equivalent to the limit of the Amitsur complex for the map $S\to MU$, but $S\to MU$ is not an effective descent morphism..... is it??

@JonathanBeardsley asking that the Tot-tower of the cobar complex is constant as a pro-object is stronger than asking that the limit of the cobar complex is A

the latter is saying that A is B-complete, while the former is saying, e.g., that the associated bousfield-kan spectral sequence has a horizontal vanishing line at a finite page

@skd ah i see, okay. yeah i noticed that it was being asked for that to be an equivalence of pro-objects, rather than just of limits, and was worried I was missing something.

so S -> MU is not a universal descent morphism

Okay good, that's a relief!

Thanks for clarifying that. :0)

I mean to do a regular smiley, like :-), but I kinda like the one with the big nose.

(but, e.g., the map L_n S -> E(n) is a universal descent morphism; this is the smashing localization thm)

haha

what's up @ReubenStern

@skd Right. Isn't there some well known counter-example though that shows that not every Galois extension (in the sense of Rognes) is a universal descent morphism?

idk if it's well known but this document gives an example folk.uio.no/rognes/papers/unfaithful.pdf

faithfulness of a galois extension is necessary (and sufficient) for the galois extension to be a universal descent morphism

btw i think some of this stuff is in arxiv.org/abs/1404.2156 but i haven't read it

@skd What does "$B$-complete" mean in this context? Is this a bousfield localization sort of term? I only know about local objects and acyclic objects, what does "complete" mean?

@skd I think "$B$-complete" is supposed to mean in the sense of "nilpotent $B$-completion", i.e., the inverse limit of the Adams tower.

yeah, that's what I meant. @CharlesRezk I think you meant to tag @SaalHardali

it's related to bousfield localization; in many cases the B-completion will be bousfield localization wrt B

the monoid axiom on cofibrantly generated symmetric monoidal model categories C provides its category of monoids with a model structure created in C. is it a monoidal model category?

@skd If S--->B is a n E_infty algebra is the statement that B-localization is smashing equivalent to the map being a universal descent morphism?

Or did you mean that the implication is only in one way

yes, if the map L_B S -> B is a universal descent morphism if B-localization is smashing

idk how to edit this on my phone, but in order for that sentence to parse you should remove the first "if"

@skd for future reference, menu button in the bottom left ~> edit last. you can't edit anything other than your most recent message without going to the desktop site

ugh sorry the implication is "L_B S -> B is universal descent" => "B-localization is smashing"

ah I see thanks

there's no menu on the bottom left

the idea is that pro-constancy allows you to move smash products past the Tot (which is a limit)

@skd oh i see, they redesigned the chat a couple years ago and i opted out so i'm probably seeing something different. my bad.

oh well. thanks anyway

Ah, in the new one you click on a message and it pops up the things you can do with it up top, including a pencil icon for messages in the past two minutes. That seems convenient. Maybe I should stop opting out

Sorry to keep interrupting the actual homotopy theory

oh I see, that's good to know

actually I'm pretty sure that B-localization being smashing is equiv. to L-B S -> B being a universal descent morphism

there is a time limit on the ability to edit. also often mobile interfaces are different and more limited

to reply to my previous question: not necessarily mathoverflow.net/a/195194/6249 (and I knew it a year and a half ago!)

an interesting thought is that, unless I'm making a very silly mistake, if the commutative monoids have a model structure, then it is monoidal model, because the pushout-product map is an isomorphism (by a "pushouts commute with pushouts" argument and the fact that pushouts in commutative monoids are relative tensor products)

ohhhhh jeeez : kerodon.net