Abelian quotients of triangulated categories

We study abelian quotient categories A=T/JA=T/J, where TT is a triangulated category and JJ is an ideal of TT. Under the assumption that the quotient functor is cohomological we show that it is representable and give an explicit description of the functor. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when JJ gives rise to a cluster-tilting subcategory of TT. We show that the quotient functor preserves the AR-structure. As an application we show that if TT is a finite 2-Calabi–Yau category, then with very few exceptions JJ is a cluster-tilting subcategory of TT.