Skip to main content
MathOverflow

You are not logged in. Your edit will be placed in a queue until it is peer reviewed.

We welcome edits that make the post easier to understand and more valuable for readers. Because community members review edits, please try to make the post substantially better than how you found it, for example, by fixing grammar or adding additional resources and hyperlinks.

5
  • $\begingroup$ I think I don't understand anything from this answer xD. I don't know what spectra are (my topology knowledge is not great), and I don't know anything about $\infty$-categories either. $\endgroup$
    – Elías Guisado Villalgordo
    Commented Sep 20 at 10:15
  • 5
    $\begingroup$ Let me reinforce Neil's answer here with another example. If you ind complete the small stable module category of a finite group as a triangulated category, you get the big stable module category modulo phantoms, which is (probably) not triangulated, and not really what you want. However, if you regard it as a stable infinity category and ind complete, you get the big stable module category that you wanted, as a stable infinity category. An intermediate solution is to use differential graded categories, and this works well, but comes with its own problems. $\endgroup$
    – Dave Benson
    Commented Sep 20 at 10:31
  • 1
    $\begingroup$ Now I am confused: Isn't the homotopy category of a stable $\infty$-category triangulated? I guess, the Ind-completion does not commute with taking the homotopy category? $\endgroup$
    – Ulrich Pennig
    Commented Sep 20 at 11:31
  • 5
    $\begingroup$ @UlrichPennig Yes it is; no it doesn't. $\endgroup$
    – Dave Benson
    Commented Sep 20 at 11:37
  • 2
    $\begingroup$ There is a short argument in Remark 5.9 of math.uni-bielefeld.de/~hkrause/completion.pdf, which I believe shows that the ind-completion (in the triangulated sense) of the stable module category is not triangulated if the Sylow p-subgroup of $G$ is not cyclic. $\endgroup$
    – Drew Heard
    Commented Sep 24 at 17:21