3
$\begingroup$

Let $E$ be a spectrum (with corresponding homology theory denoted $E_\ast$).

In "Localization of spaces with respect to homology", Bousfield constructed a model category structure on the category of based spaces where weak equivalences are detected by $E_\ast$-homology.

In general, the $E$-local model structure is not right proper, e.g., for $E = H\mathbb{Q}$ (i.e. $E_\ast = H_\ast(-,\mathbb{Q})$), Quillen provides a counter-example to this in "Rational Homotopy Theory".

I'm interested in compiling a list of spectra (homology theories) $E$, for which the $E$-local model structure is right proper.

Any references/proofs would be greatly appreciated.

Improve this question
$\endgroup$
4
  • $\begingroup$ Do you know Bousfield's 2000 paper "On the telescopic homotopy theory of spaces"? In the appendix he proves a theorem about when a left localization is right proper. I'll bet you could use that to characterize spectra $E$ where the local model structure is right proper $\endgroup$
    – David White
    Commented Apr 15, 2021 at 18:08
  • $\begingroup$ It doesn't seem that the paper you mentioned has an appendix, at least not on any of the copies I have access to $\endgroup$
    – Niall Taggart
    Commented Apr 15, 2021 at 18:21
  • $\begingroup$ Ah, sorry, it's in Appendix A of Bousfield Friedlander, "homotopy theory of $\Gamma$ spaces, spectra, and bisimplicial sets" then improved in Section 9 of the telescopic paper I mentioned. $\endgroup$
    – David White
    Commented Apr 16, 2021 at 0:29
  • $\begingroup$ Thanks. I'm fairly familiar with both of those references, but I'll have another look to see if I can extract what I'm looking for $\endgroup$
    – Niall Taggart
    Commented Apr 16, 2021 at 8:33

0

Reset to default

You must log in to answer this question.