Homotopy Theory

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12:06 AM

Say I have a naive G-spectrum X for G finite, if the homotopy fixed point SS for X collapse at E_2, is it possible to conclude that the homotopy fixed point SS for X smash X, treating as a naive G-spectrum with diagonal action, also collapse at E_2?

Or is there any counter-examples?

William Balderrama

12:48 AM

@CharlesRezk I don't know a published reference, but it appears as Proposition 7.1.17 of Cisinski's notes: mathematik.uni-regensburg.de/cisinski/CatLR.pdf .

12 hours later…

Sean Tilson

12:39 PM

@MingcongZeng a seemingly irrelevant question: does the spectrum in question have non trivial mod p homology? Is it p-complete?

If it does and is then I might guess yes. I would daisy chain the adams SS with the homology based homotopy fixed point SS.

Achim Krause

Sean, I don't think "the Homotopy fixed point SS degenerates at E_2" and "the mod p Homology version of the Homotopy fixed point SS degenerates at E_2" are even equivalent.

Sean Tilson

I also don't think they are.

So it seems like what is desired is a sort of K\"unneth theorem result and stable homotopy does not have a good K\"unneth thm/ss.

Twistediso

1:13 PM

Let us consider the category of sheaves on some site, and say I have an abelian sheaf \mathcal{F}. Then if \mathcal{F} is actually a sheaf of say Z/n-modules, then the cohomology H^i(X,F)= Ext_Z^i(Z,F) agrees with Ext_(Z/n)^i(Z/n,F). Is there a "modern" proof of this fact? The one I have in mind relies on noting that an injective resolution of F as Z/n-modules maps to a flasque resolution as Z-modules.

Denis Nardin

1:25 PM

@Twistediso Well, it depends on what you mean by "modern" or how you define H^i(X;F). Ultimately it boils down to the fact that the forgetful functor D(Z/n)→D(Z) preserves limits, but with the definition of cohomology as an Ext I think your best bet is still to prove that Ext_Z^i(Z,-) is an effaceable \delta-functor on Z/n-modules

1 hour later…

Joe Berner

2:47 PM

if I have the Cech-local \infty-topos X on a site, are there known conditions for when the mapping space Map_X(\mathbb{1}_X, K(A,n)) computes Cech cohomology of the constant sheaf A? It is a trivial argument to show that the mapping space agrees up to equivalence with the corresponding one in the hypercompletion of X, and I'm wondering if this gives new conditions for when Cech cohomology of constant sheaves agrees with derived functor cohomology

Charles Rezk

3:18 PM

@WilliamBalderrama Thanks

6 hours later…

Arrow

9:35 PM

Hello. Forgive the basic question.

Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote the *tangent set* of $X$ at $p$, namely the set of derivatives of curves in $X$ based at $p$ and differentiable at zero.

Are the following conditions equivalent?

1. $\mathrm{T}_pX$ is a linear subspace of $\mathbb R^n$ of dimension $\dim_pX$.
2. We have the following equality, where the limit is taken in a translated neighborhood in $X$ of $p$. $$\lim_{h\to 0} \frac{\pi_{(\mathrm T_pX)^\perp}(h)}{\|…

Arrow

9:54 PM

(In the second condition just replace the tangent set with some vector subspace, so as not to assume the tangent set is linear.)

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