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Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of compact objects of $B$ (I don't need this embedding to be an equivalence; yet if some $B$ of this sort exists then one can always "shrink" it so that its subcategory of compact objects will become the Karoubi envelope of $T$)?

I would conjecture that such a category $B$ exists for any $T$; yet does there exist any way to prove this? What is the largest class of small triangulated categories for which this statement is known?

I suspect that the proof should be "known" whenever some "enhancement" exists for $T$; yet any comments or references would be very welcome!

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I do not know of an answer for a general triangulated category (non-topological triangulated categories are very unusual), but as soon as you ask for some more structure the thesis follows very quickly.

Let us suppose that $T$ is the homotopy category of some stable ∞-category $C$ (that is, that $T$ is a topological triangulated category in the sense of Schwede (arXiv:1201.0899) ). Then the idempotent completion (sometimes called the Karoubi envelope) of $C$ is the category of compact object for the stable ∞-category $Ind(C)$ by lemma 5.4.2.4 in Lurie's Higher Topos Theory, so the idempotent completion of $T$ is the category of compact objects of the homotopy category of $Ind(C)$.

The fact that $Ind(C)$ is stable (and so its homotopy category is triangulated) is proposition 1.1.3.6 in Lurie's Higher Algebra.

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  • $\begingroup$ Thank you! May I ask you (sorry for my ignorance): can all triangulated categories that admit some sort of an "enhancement" be presented as stable $ \infty$ categories? $\endgroup$
    – Mikhail Bondarko
    Commented Nov 23, 2016 at 18:36
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    $\begingroup$ @MikhailBondarko I cannot vouch for all sort of "enhancements", but a stable ∞-category is, roughly speaking, a triangulated category enriched in spectra; so all dg-enhancements should be special cases of it (e.g. arxiv.org/abs/1308.2587). $\endgroup$
    – Denis Nardin
    Commented Nov 23, 2016 at 18:38
  • $\begingroup$ @MikhailBondarko Another reference that might be useful is Construction 1.3.1.6 in Higher Algebra, that for every dg-category constructs an ∞-category with the same homotopy category. I believe that when the original dg-category was triangulated this ∞-category will be automatically stable. $\endgroup$
    – Denis Nardin
    Commented Nov 23, 2016 at 18:46
  • $\begingroup$ Thank you! So, you essentially confirm my guess. I wonder whether there exist arguments that are quite distinct from "take a enhancement for $T$ and consider ind-objects for it". $\endgroup$
    – Mikhail Bondarko
    Commented Nov 23, 2016 at 18:51
  • $\begingroup$ @MikhailBondarko I very much doubt that a given triangulated category has a stable $\infty$ category associated to it. The example of Muro-Schwede-Strickland "Triangulated categories without models" should be a counterexample. I believe the obstructions they find are present even on the $\infty$-category level, not just the model category level. $\endgroup$
    – David White
    Commented Nov 23, 2016 at 20:04

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