# Can triangulated categories be “approximated by countable subcategories” (that are triangulated but not full!)?

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.

• Each instance of every axiom says that, given some finite set of objects and morphisms, there is a finite set of objects and morphisms satisfying some property. So starting with a countable set of objects and morphisms, can you not just add countably many objects and morphisms so that all instances of axioms whose hypotheses involve only the original set are satisfied? And then you can iterate and take the union, to get a countable triangulated subcategory. – Jeremy Rickard Jun 28 '15 at 10:42
• More strongly, all of the structure and axioms of a triangulated category can be expressed in first-order logic with a finite language, so by Lowenheim-Skolem given any set $S$ of morphisms, there is a subcategory of size at most $\aleph_0+|S|$ containing $S$ which is not just triangulated but an elementary submodel of $C$. – Eric Wofsey Jun 28 '15 at 12:45
• Yes, such a logical argument should work. Yet I would prefer to avoid it since it is not quite "inner mathematical". – Mikhail Bondarko Jun 28 '15 at 16:56
• @EricWofsey Of course, the usefulness of this observation is directly proportional to the expressiveness of the first-order language in question. But I am sure you know that. – Zhen Lin Jun 29 '15 at 15:23
• @ZhenLin: Yes, of course. Indeed, I suspect that not very many interesting properties of triangulated categories are first-order. Still, I wanted to give some indication of how far the idea in Jeremy Rickard's comment could be pushed. – Eric Wofsey Jun 29 '15 at 15:30