Homotopy Theory

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Tomer Schlank

06:51

@DenisNardin @MarcHoyois and @YonatanHarpaz, Thanks!!
I think there is a slight problem with the reduction to the mapping space argument which is that using DK correspondence gives simplicial sets but If I'm not mistaken does not give simplicial model category. However HA 1.3.4.5 seems to be almost what I need (There, one need to restrict to "fibrant-cofibrant objects") but maybe I can push it trough .

D1811994

07:39

@DenisNardin I come because I still have doubts about the question http://mathoverflow.net/questions/253069/invariance-of-combinatorial-geometric-euler-characteristic. We had been discussing whether or not the spaces were locally compact. One of that spaces could be 2-simplex with and edge (boundary face) removed (here I have made a picture of it: https://drive.google.com/file/d/0B5QLY6KP0JyvdW9RUGxRVm5meDA/view?usp=sharing). I think that is not locally compact.

And I don't know if Borel-Moore Homology can be used with spaces which are not locally compact.

5 hours later…

Denis Nardin

12:31

@D1811994 Hmm it seems that you are right. I am pretty sure that the Borel-Moore homology I described works also in this case (where maybe you need to be a little bit careful and define a locally finite chain as a chain that is finite in the nbd of every point) but I don't know of a reference

D1811994

13:03

@DenisNardin Thank you very much anyway!

2 hours later…

Denis Nardin

15:15

This question might be a bit too broad but: what is known about the $\pi_0$ of suspension spectra of varieties in $SH(S)$ (stable motivic category)? Is there anything known besides the Hopkins-Morel theorem?

Marc Hoyois

16:14

@DenisNardin If S is a field, they're oriented Chow group of 0-cycles: arxiv.org/abs/1108.3854v1

Espen Nielsen

Reality check: I have a pair of symmetric monoidal functors $F,G:C\rightarrow D$, for dg-categories. There is a natural homotopy equivalence $\eta:F\rightarrow G$. If $\eta$ is a symmetric monoidal transformation, then any natural homotopy inverse is an $E_\infty$-monoidal transformation, right?

Sean Tilson

Dumb question, but how do you get from Hopkins-Morel to your question @DenisNardin?

(I am happy for the any to answer obviously.)

Denis Nardin

@MarcHoyois Thank you!

@SeanTilson Well when X is a point, $\Sigma^∞_+X$ is the sphere spectrum and Hopkins-Morel tells you its $\pi_0$

Sean Tilson

Thanks.

That sure was a dumb question. I guess I always think that the Hopkins-Morel theorom is something else (the identifying the 0-slice of MGL which I guess is a corollary? Is that right?).

2 hours later…