# On two notions of 'generators' for a 'large' triangulated category

Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if

(i) There are no proper triangulated subcategories of $C$ that are closed with respect to small coproducts and contain $D$.

or if

(ii) For any non-zero object $c$ of $C$ there exist an integer $i$, a $d\in D$, and a non-zero $C$-morphism from $d[i]$ to $c$.

My question is: are there any 'well-known' relations between conditions (i) and (ii)? I only know a proof of their equivalence in the case when $D$ is a set of compact objects.

The obvious implication is (i) $\Longrightarrow$ (ii), of course. Indeed, the class of all $c\in C$ not satisfying (ii) is precisely the full triangulated subcategory of objects right orthogonal to the minimal triangulated subcategory satisfying (i) in $C$. Any triangulated subcategory with a nontrivial orthogonal subcategory is proper, hence the negation of (ii) implies the negation of (i).
• The argument goes like this: suppose that the Brown representability holds for the minimal triangulated subcategory $T$ in $C$ satisfying (i). Given an object $x\in C$, consider the functor $Hom_C({-},x)$ on the category $T$ and denote by $t\in T$ an object representing it. Then the identity morphism $t\to t$ corresponds to a certain morphism $t\to x$. The cone $c$ of this morphism satisfies the negation of (ii), so it must be zero. – Leonid Positselski Jul 25 '14 at 13:53