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Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and reflective?

Thanks, Jon

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    $\begingroup$ Do you mean in the sense of Hovey, Palmieri, and Strickland's book? I'm pretty sure that this is equivalent to saying that the subcategory is closed under colimits. (A useful reference here is 1.4.4.1 of Lurie's "Higher Algebra.") The point is that once your subcategory has all coproducts and is stable, you get all colimits for free. $\endgroup$
    – Akhil Mathew
    Commented Nov 15, 2012 at 1:09
  • $\begingroup$ Yes I do. Thanks, I will check out Lurie's book. $\endgroup$
    – Jonathan Beardsley
    Commented Nov 15, 2012 at 1:50
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    $\begingroup$ Your question rises a nice point, ¿are all localizing subcategories of a triangulated categories reflective? I think that, so far, the answer is only known 'up to set theory'. $\endgroup$
    – Fernando Muro
    Commented Nov 15, 2012 at 8:59
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    $\begingroup$ Yeah, that is an interesting question to me as well Fernando. However, @Akhil, I feel that maybe being closed under all colimits is too strong a condition? $\endgroup$
    – Jonathan Beardsley
    Commented Nov 15, 2012 at 16:45

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