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Suppose given a well-generated triangulated category with a compatible symmetric monoidal structure, $\mathcal{T}$ (in the sense of Neeman). Is it clear that the image of a localization functor will also be well-generated? If not in true in general, is it easy to prove with respect to the $X$-acyclics (i.e. objects which tensor with $X$ to zero) for some element $X$?

The main thing I'm worried about I suppose (though the second axiom for well-generated-ness is not obvious to me either) is the requirement for the generators to be $\alpha$-small for some cardinal $\alpha$. For instance, the $MU$-localization of the stable homotopy category has no small objects. But perhaps they are $\alpha$-small for some $\alpha$? My understanding of such issues is still somewhat superficial.

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The quotient of a well generated category by a localizing subcategory generated by a set is again well generated. All localizing subcategories are generated by sets if we assume some set-theoretical axioms. – Fernando Muro Oct 2 '12 at 22:21
Thanks @Fernando! Is this somewhere in Neeman's book? Also, do you happen to know which set-theoretic principles must be assumed? – Jon Beardsley Oct 2 '12 at 22:28
Why does the $MU$-localization of the stable homotopy category have no small objects? – Akhil Mathew Oct 3 '12 at 11:28
Hey @Akhil I don't really know why, but the proof/explanation is in appendix B (specifically Cor. B.13) of Hovey and Strickland's "Morava K-theoriesand Localisation" which I believe can be found here: This is also discussed briefly in Example 3.5.4(a) of Hovey, Palmieri and Strickland's "Axiomatic Stable Homotopy Theory" (…) although I'm sure there are people on this site who can explain it perhaps intuitively. – Jon Beardsley Oct 3 '12 at 12:58
@Akhil no problem! I'm pretty sure I was sitting behind you at the Dan Quillen memorial conference! :-) – Jon Beardsley Oct 13 '12 at 3:12

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