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! 



Which triangulated categories are subcategories of compact objects "somewhere"?
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](https://arxiv.org/abs/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*.


