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Peter Haine
Peter Haine
12:58 AM
@PiotrPstrągowski I also think that's how the left adjoint α* is supposed to be defined. In the case that S is a profinite set, I think there is a more straightforward way to proceed. Since quasicompact open sets form a basis for the topology on S, the topos Sh(S) can be described as sheaves on the site of compact opens of S. Then I think the obvious functor {compact opens of S} → Comp/S is a morphism of sites and gives rise to the desired geometric morphism.
(I didn't check this carefully though, so take it with a grain of salt.)
Also, sorry I used both "quasicompact" and "compact" to mean the same thing. I've been in the habit of using "quasicompact" more often, but when I edited to change the message to use "compact" I missed one of them and I can't seem to edit the message again…
 
 
4 hours later…
Yuri Sulyma
Yuri Sulyma
5:31 AM
@DylanWilson nice paper! (@Jeremy and @Andy too, but they don't seen to be on here.) Does anything go wrong if I take the naive generalization to define $\mathbb{CP}^\infty_{\mu_n}$? (or $\mathbb{CP}^\infty_{\mu_{C_{p^n}}}$) I'm pretty sure $\mathrm{THH}(R;\mathbb Z_p)$ is $C_n$-homotopically even for all $n$ (I haven't checked yet but it's easy to do so)
 
 
3 hours later…
Dustin Clausen
Dustin Clausen
8:02 AM
Hi Piotr! Yes, that's the map of topoi. You can also think of it like this: given a sheaf on S for the usual topology and a map of compact hausdorff spaces T --> S, just pull back the sheaf and take global sections on T. This describes the "pullback" functor part of the geometric morphism of topoi. (Sometimes it's best to just forget about sites.)
The more general version of 3.2 is that the associatation S \mapsto (sheaves of spaces on the topological spaces S) for compact hausdorff S is itself a sheaf of \infty-categories for the topology of finitely many surjective maps. And this is some usual "proper descent" result which follows from proper base change.
(for some reason Peter decided to basically just reprove proper base change in the argument he wrote up for 3.2)
Sorry, maybe I should make it clear why that's a generalization of 3.2. The point is that you want to compare the above sheaf of \infty-categories with the other one S \mapsto (sheaves of spaces on compact hausdorffs over S) which satisfies that same descent more-or-less tautologically. Namely you want to see the first is fully faithful in the second (on bounded above sheaves, say, to avoid hyperdescent issues). The descent reduces you to extr. disc. spaces where it's clear because
in both cases covers are refined by disjoint open decompositions
Okay, that was probably more tha you were asking for :) i just had the impression that for \infty-people there was a way of rephrasing what Peter's doing which might be more familiar
also here's a case where i probably should have said "anima" instead of "spaces" to avoid confusion...
 
 
2 hours later…
Igor Sikora
Igor Sikora
10:35 AM
Hi all, I need some proper reference for the following problem - if I have RO(C_2)-graded homotopy groups of some commutative ring spectrum, how do I know that these are commutative as a ring?
Specifically, I know that there is some business with units in Burnside ring, but Lewis-Mandell's description of that is rather sketchy - so I need some other reference
 
Igor Sikora
Igor Sikora
10:51 AM
Please ignore the above - I was looking at wrong part of the paper
 
Dylan Wilson
Dylan Wilson
@IgorSikora You need to check that [C_2]=2 in pi_0, i.e. that the transfer of 1 is 2.
 
Igor Sikora
Igor Sikora
@DylanWilson And that gives me ordinary (not graded) commutativity?
 
Dylan Wilson
Dylan Wilson
oh no, sorry, that would give you graded commutativity
I don't think it's possible to get 'ordinary' commutativity
for example, the integer stems would be graded commutative
(unless 2=0, I guess, or you're concentrated in even degrees)
 
Igor Sikora
Igor Sikora
Yup, but I forgot that I'm working with a ring where nearly all stems are Z/2, so I'm good - thanks!
Do you have any other reference for that then Lewis-Mandell?
 
Dylan Wilson
Dylan Wilson
@YuriSulyma Thank you! I guess I could believe that the space ''$\mathbb{C}P^{\infty}_{\mu_n}$'' looks even when n is not prime, but I think the connection with chromatic homotopy theory is less clear. That is, I'm not sure if I would expect Morava E-theories to be oriented in the analogous way. A clue for this is that these spaces are privileging the (integral) reduced regular representation in their underlying homotopy groups, but, in general, you'd want to find
some integral version of the action of finite subgroups of the Morava stabilizer group on the Dieudonne module of your formal gr…
 
Igor Sikora
Igor Sikora
11:32 AM
Thanks a lot!
 
 
11 hours later…
Piotr Pstrągowski
Piotr Pstrągowski
10:30 PM
@DustinClausen Thanks, this is great! Took me a second to understand this, but I think I see the point now.
This must be standard for people who think about this, but it seems to me proper base change theorem tells you that the functor $S \rightarrow Sh_{open}(S)$ (covariant, so pushforward of sheaves) preserves pullback. From this I can deduce that after applying $Sh_{open}$ to the Cech nerve of a map $S^{\prime} \rightarrow S$ of compact Hausdorff spaces, I'll get the Cech nerve of $Sh_{open}(S^{\prime}) \rightarrow Sh_{open}(S)$.
If now I knew the latter is an effective epimorphism in $RTop$, then I would deduce that the "left-adjoint dual" of this diagram is a limit diagram in $LTop$, which is what we need.
How would I see that this is the case if $S^{\prime} \rightarrow S$ is a surjection? Or is there a different way to deduce that $Sh_{open}$ satisfies descent once you know proper base change? (I apologize, I imagine this must be standard, but I couldn't figure it out after looking through HTT.)
 
Jonathan Beardsley
Jonathan Beardsley
10:52 PM
Anyone know what the compact objects in sSet are? Or in the underlying ∞-category?
 
Piotr Pstrągowski
Piotr Pstrągowski
@Jonathan In the category of simplicial sets, these are the simplicial sets with finitely many cells.
 
Jonathan Beardsley
Jonathan Beardsley
@PiotrPstrągowski Ah okay, great. At one point I knew this.
 
Piotr Pstrągowski
Piotr Pstrągowski
For the second question, what do you mean by the underlying $\infty$-category? With the Quillen model structure?
 
Jonathan Beardsley
Jonathan Beardsley
Eh, don't worry about the second question.
Oh also @blank_space I just read through all that stuff. I basically feel the same way, though since I now I have a tenure-track job that I like, I get that I can't really complain. You're also welcome to e-mail me, if you ever wanna chat about math, or math culture, or whatever else.
My main strategy has just been to constantly ask "easy" questions, and constantly have smart people answer them. I often feel that I'm not really a mathematician since so much of my work has depended on me being able to get help from people who know a lot of technical details really well. This more or less dogs me every time I try to do mathematics.
I attempt to be somewhat honest about this (without being uncomfortably self-deprecating) when I write papers by thanking people who have helped me figure things out.
So my acknowledgement sections are often quite busy, I think...
Anyway... I feel you. And I appreciate you sharing your experience. I think compared to things like, I don't know, combinatorics, or low-dimensional topology, homotopy theory can be pretty brutal and restricted and inaccessible (although I might be biased here, and would be happy to be told to shut my trap by a combinatorialist or a knot theorist) in the sense that it seems to only be "happening" at "fancy" places (although maybe that's changing).
But yeah, on the other hand, I had a famous advisor, went to three Talbots, several YTMs, and hang out in this chat room all the time, so I'm pretty "connected."
(which by the way contributes further to my very strong feeling that I'm not actually all that good at math and have just been able to leverage being friends with a lot of extremely talented people)
 

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