8:21 AM
Here's a naive question: Whats the relationship between the connectivity of a map and connectivity of its cofiber? In the case where all fundamental groups are zero I know how to prove (using hurevitz and serre ss) that if the cofiber is $n$-connective then the map is $n$-connective (maybe upto a constant minus/plus 1). I'm wondering about the general case, is it true without restriction on $\pi_1$?

4 hours later…
12:41 PM
Just in case somebody knows and is willing to help. (I have to admit that I know almost nothing about 2-categories, bicategories and similar stuff.)
in Category Theory, 1 min ago, by mattecapu
Hi folks, does someone know, in the context of weak 2-categories, what's the standard terminology to distinguish between
(a) an invertible 2-morphism
(b) a 1-morphism which is invertible up to an invertible 2-morphism (like an homotopy equivalence)
As of now I'm calling (a) iso-2-morphism and (b) 2-isomorphism, which is ambiguous

1:38 PM
If I may come back to my issue with equivariant cohomotopy of the point in non-trivial RO(G)-degree, on the off chance some equivariant homotopy theorist here could offer some help:
I have most of the proof that the G-equivariant cohomotopy of the points in some RO(G)-degree V is an extension of the Burnside ring A(G/N) whenever N is a normal subgroup of G such that V^N = 0.
I have written this out here:
But currently I use one step in this proof which looked trivial on first sight, but really requires an argument.
So there remains a gap.
I am still hoping this gap may be bridged by an abstract argument, because it looks so very obvious. If not, it may require diving into Segal's proof that in degree 0 we have exactly the Burnside ring...

2:25 PM
@UrsSchreiber The G-set S in A(G) is the trace of the strongly dualizable spectrum Sigma^infty S_+; this is intrinsically defined and preserved by any symmetric monoidal functor (as you probably know better than I do). Consequently the map A(G) -> A(G/N) induced by the symmetric monoidal partial geometric fixed points functor Phi^N must send S to the trace of Phi^N(Sigma^infty S_+). Since Phi commutes with Sigma^infty, your claim follows.

3 hours later…
5:37 PM
@TomBachmann thanks for offering help! Sorry for being slow, but I need to think about this: 1)How do we see that the comparison map to the partial fixed point spectrum is monoidal? and 2) Why is the G-set that trace?

5:47 PM
@UrsSchreiber (1) That the geometric fixed point functor is symmetric monoidal is a classical fact (proven, e.g., in Hill-Hopkins-Ravenel). One way of seeing this is that it is the localization induced by the idempotent object $Σ^∞\widetilde{EP_N}$ where $P_N$ is the family of subgroups not containing N
(2) follows from the identification of the unit and counit morphisms for the self dual spectrum S (plus the tom Dieck splitting, ofc).

A very high-powered way of proving (1) is also Remark 9.9 here: arxiv.org/pdf/1711.03061.pdf

1 hour later…
7:07 PM
Thanks. I am aware that the geometric fixed point functor is symmetric monoidal, but what I'd seem to need here is that the comparison map from E-cohomology to Phi^N E-cohomology to be symmetric monoidal, no?

I'm not sure I follow. The two spectra don't even live in the same category
Here by "trace" Tom means the thing that associates to a (strictly) dualizable object X of a sm category the composition $tr(X):=[1→X^\vee\wedge X→1]∊[1,1]$
So every time you have a symmetric monoidal functor F you get the identity $F(tr(X))=tr(F(X))$

@DenisNardin the cohomologies though are both spectra or abelian groups. Explicitly, I am concerned with this map right here: ncatlab.org/nlab/show/geometric+fixed+point+spectrum#Hmm

I still do not follow. The map A(G)→A(G/N) is literally the map $[1,1]_{Sp^G}→[1,1]_{Sp^{G/N}}$ induced by $\Phi^N$
Tom's description of [S] as the trace of Σ^∞_+S shows that this map sends [S] to [S^N] (as we would expect)

7:25 PM
Thanks. In that proof I was trying to complete, the map A(G) --> A(G/N), does not manifestly arise this way, instead it arises as a component map of the colimit in Prop. II 9.13 IN Lewis-May-Steinberger.
Maybe I should abandon that whole proof strategy, but this is in any case what I was asking about: Why is that component projection behind the link I gave equivalent to that evident/canonical map?

Oh, but that's pretty much the same thing
You can identify the two things if you identify $\Phi^N$ with the smashing localization associated to the idempotent $Σ^∞\widetilde{EP_n}$
Since you can show that $EP_n$ is the colimit of all those S^V's
(the argument for that is in the proof of II.9.13 in LMS, where they call what I call $\widetilde{EP_N}$ $\tilde E\mathcal{F}_N$

Thanks! That helps. The statement about smashing localization is secretly what they have around Prop. Ii 9.4, I suppose.
I expect it would all be pretty trival if there were a 2018-analog of Lewis-May-Steinberger, and probably that's exactly what exists internal to you guys.
My task is to write out a proof that can be recognized by readers who are less than homotopy experts...
I will have to call it quits for tonight, thanks a lot for the chat.
But while we are at it, allow me to ask: I just meant to prove here that S_G^V(pt) --> S^0_G/N(pt) is surjective when V^N = 0. But ideally I'd like to know S^V_G(pt) completely. What can one say?

7:45 PM
So, thanks to the tom Dieck splitting $(Σ^∞S^V)^G=\bigoplus_{(H)} (Σ^∞S^{V^H})_{hWH}$, so the group you're asking for is \$\bigoplus_{(H)}π_0(S^{V^H})_{hWH}
(I'm assuming the group is finite for the moment 'cause I can never remember how the shift go in the general case)
So, I guess the question is, for which H is (S^{V^H})_{hWH} connected?
Note that the homotopy quotient is taken in pointed spaces