For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them. Certainly, there should not exist "the triangulated subcategory generated by $f_i$"; yet I suspect that one choose a $C'$ such that its objects and morphism set is countable (in the case when my set of $f_i$ is countable). Are any results of this sort known?
Upd. Possibly this information would be interesting to somebody: we have written http://arxiv.org/abs/1508.04427; section 3 in it is mostly dedicated to the relation of triangulated categories to their countable subcategories.