12:25 AM
@TylerLawson i'm trying to organize my ''''research program'''' and record questions i've (sometimes briefly) thought about in a way that's also conducive to selective collaboration
i'm sure to share a link in here when it's closer to finished. it's functioning, but the interface isn't what i want it to be
@Adeel i've emailed you quillen's notes, let me know if it didn't come through
@SeanTilson if we're talking about the folk etymology, BP<n> is a more sensible delocalized object to take

1 hour later…
1:37 AM
Does anyone have any intuition at all for this LCut functor that Lurie produces in 4.2.2.6 of Higher Algebra?
It should be doing something like picking out the necessary module structure parameterized by his $LM^\otimes$-operad.

2 hours later…
3:46 AM
What happens if I try to talk about strictly associative multiplications in $\infty$-categories? Will my face melt off or something?

4:25 AM
nothing horrific will ensue
you can take associative monoids in the 1-category of quasicategories, or you can take A_oo monoids in the oo-category of quasicategories
you're doing essentially the same thing in each case - those two things are equivalent homotopy theories for the same reason as A_oo-spaces are equivalent to strictly associative H-spaces

4:45 AM
ah good.
I sort of misspoke (mistyped?) up there. I don't really want strictly monoidal $\infty$-categories, I just want to talk about a strict monoid in an $\infty$-category.

how might one define such a thing?

Well, give an object, with a map $X\times X\to X$ and then say that the associativity diagram commutes strictly.
Perhaps it'd be better to say this inside of a topologically enriched category and then take the simplicial nerve or something.
In other words $\mu(\mu\times 1)=\mu(1\times\mu)$

right, it's not meaningful to say the a diagram commutes strictly in a quasicategory

Yeah.
I didn't think so.

you can talk about a strictly commuting diagram in a topological category, but after you take the simplicial nerve there's no trace of that strictness anymore

4:50 AM
That's fine with me, honestly. All I need is for something to be $A_\infty$, haha, but it's much easier to show that it's strictly associative.
90% of my postdoctoral work will consist of going back and answering all of my MathOverflow questions.

5:24 AM
having said that, I think there should be a homotopy invariant notion of strict commutativity
but that's sort of conceptually tricky territory, for me at least

@SaulGlasman I've played with this a little bit, and Lurie makes a remark somewhere in his thesis about it. It's actually not so hard to talk about in the cartesian monoidal case. I'm not quite sure what to do elsewhere, but would very much like to.

6:04 AM
this is that thing about, in a cartesian infinity-category C, equipping an E-infinity object A with a lift of Hom(-, A) : C -> connective spectra to a functor C -> connective HZ-modules (or something like that)?

10 hours later…
3:40 PM
You can talk about it that way, or you can talk about the fact that you have honest symmetric power functors in this setting, so you can write down an actual monad encoding these things

4:25 PM
Mike Hill + Vitaly Lorman = homotopy fixed point spectral sequences all over these boards

1 hour later…
5:51 PM
Suppose I take an $\infty$-groupoid's geometric realization $|X|$, and then take the fundamental $\infty$-groupoid of that, do I get an equivalence (in Kan complexes) $\Pi_{\infty}(|X|)\simeq X$?
This seems like it should really be true.

Yes it is true
If I understand your notation you get a canonical map $X\to \Pi_\infty(|X|)$ and this is a weak equivalence

Right, the unit of this adjunction?

yes, this is more or less the fact that $|-|$ and $\Pi_\infty$ is a Quillen equivalence between the Kan model structure and the Quillen model structure
So the adjoint of the identity must be a weak equivalence

Yeah, okay. I thought it was something like that. Perhaps I should just be using $Sing(-)$ anyway

Maybe I'm confused: what's the difference between $\Pi_\infty$ and $Sing(-)$? I thought they were the same thing

5:55 PM
I just meant, like, notationally.

Ah ok

I was just looking at nlab
So, I had $\Pi_\infty$ in my brain.
Oh, Denis, as long as you're here - suppose I want to work with a topologically enriched version of simplicial presheaves on an $\infty$-groupoid $Fun(K,Spaces)$, any idea what's the right notion? Something like compactly generated weak Hausdorff spaces over $|K|$?
(In other words, I just want to work in a topologically (or simplicially) enriched category, not a quasicategory)

(sorry I'm stupid, you wrote it)

yeah, Kan complex

I think you may want Serre fibrations over $|K|$
If you'd be as happy with a Kan-enriched category you could take left fibrations over $K$

5:59 PM
Ah yeah. Or perhaps that comes from a model structure on $Top_{/|K|}$?

Yeah, my intuition is that if you want the correct mapping spaces you want your objects to be fibrant (and cofibrant but that's easy)

You mean left fibrations in $sSet$?
Or in $Kan$?

In sSet
You can take as mapping space the interior of the "obvious" mapping space $Fun_K(-,-)$
and this is a category enriched in $Kan$ modelling $Fun(K,Spaces)$ by straightening-unstraightening

Yeah. I'm not sure what exactly I'd be "happy" with. Like I was saying in here earlier, all I really need is a setting in which it's easy to show that a certain structure is strictly associative.
And then apply $N(-)$ to produce an $A_\infty$-object in $Fun(K,Spaces)$

My intuition is that whatever you are able to do with spaces over $|K|$ you can do more easily with left fibrations over $K$, but I have been wrong before

2 hours later…
8:05 PM
@DenisNardin are the fibrant objects in some model structure on $sSet_{K}$ the left fibrations?
Or I guess equivalently, is there some model structure on $sSet$ in which the left fibrations are the fibrations?
oh this is what lurie calls the "covariant model structure"

I've just discovered that left fibrations with a basis a Kan complex are Kan fibrations

Ah. Yes, I think I recall seeing this somewhere...

So I suppose you could take the natural model structure on the overcategory (I'm pretty sure there's one)

There is.
I mean, in other words, I could just take the Quillen model structure on sSet, if I want the fibrant objects of sSet_{/K} to be the left fibrations X-->K

I think you can cook up a model structure even when $K$ is not a Kan complex, but I don't know of any reference

8:10 PM
There's a standard way of going from "model structure on C" to "model structure on C over X in C"

if i remember correctly though, don't you use a different model structure, not the usual over-category model structure?

For what?

to present left fibrations

found it: proposition 2.1.4.9 in HTT

Well, Lurie shows that there's this model structure on sSet called the covariant model structure such that the overcategory presents left fibrations as fibrant objects
i think

8:11 PM
right, but it's not the model structure induced by the Joyal model structure on sSet

Certainly not.
And yeah, that nlab article identifies this as the "covariant model structure" later on.

right sorry, as you were

Just another name for it.
But, since, as Denis says, a left fibration over a Kan complex is always a Kan fibration, we can equivalently take the overcategory in the Quillen model structure if our base is Kan
Or, actually, would that give us too many fibrant objects?
No, I think it's fine.
Yay, model categories!

...argh.
If E is complex oriented and E^* is p-torsion free, then the E-cohomology of BZ/p is E^*[[x]]/[p]_F(x) and the transfer ideal is generated by the divided p-series <p>_F(x) = [p]_F(x) / x.
However, I'm having a lot of trouble locating a reference that doesn't specialize to the case of E-theory, or a coefficient ring which is complete local, or which does the Tate cohomology calculation but not this calculation.
Does anybody know one off-hand? (@Nat?)

9:20 PM
@TylerLawson I'm not sure how you would conclude that [p]_F(x) isn't a zero divisor without Weierstrass preparation.

@TylerLawson Quillen's MU paper?

10:18 PM
@DenisNardin ah i just remembered why I want to work within some honest category of topological spaces, b/c i'm working with an h-space and i want to replace it with a space with a strict unit, and i'm not sure i can do that for, uh, h-simplicial sets, haha.
Erg, I'm very confused. Is the Sing(-), |-| adjunction a quillen equivalence between sSet (with the Quillen model structure) and the category of compactly generated hausdorff spaces (with quillen model structure)? Or some other category of topological spaces?

10:41 PM
the one I'm familiar with is for CGHaus, but in principle you could substitute other reasonable full subcategories

Yeah, I think you can use this Kelley space thing. But... yeah, CGHaus works.

Hirschhorn seems to say that Quillen constructed a model structure on all topological spaces
but anyway, if you want geometric realisation to preserve finite products, you need some kind of compact-generation condition

Right, yeah, Hovey says the same.

11:33 PM
naive question
is the dual steenrod algebra free over the dyer-lashof algebra? I know it should be possible to deduce the answer to this from Steinberger's thing in the H_\infty volume, but I haven't been able to

11:50 PM
@PeterNelson for a super silly reason: if [p]_F(x) = px + higher order and g(x) = ax^k + higher order, then their product is (p*a)x^{k+1} + higher order, so if p isn't a zero divisor this leading coefficient can't be zero

derp. anyway, if you know that, then the proof in HKR needs nothing else.

@SaulGlasman as a module, no (the dyer-lashof operations annihilate 1) and as an algebra, also no (at p=2, there's a relation Q^3 \xi_1 + Q^2 Q^1 \xi_1 = \xi_1^4 + \xi_1^4 = 0).