Filtrations in abelian categories with a tilting object of homological dimension two

We consider filtrations of objects in an abelian category $\text{catA}$ induced by a tilting object $T$ of homological dimension at most two. We define three disjoint subcategories with no maps between them in one direction, such that each object has a unique filtation with factors in these categories. This filtration coincides with the the classical two-step filtration induced by torsion pairs in dimension one. We also give a refined filtration, using the derived equivalence between the derived categories of $\text{catA}$ and the module category of $En{d}_{\text{catA}}(T{)}^{op}$. The factors of this filtration consist of kernel and cokernels of maps between objects which are quasi-isomorphic to shifts of $En{d}_{\text{catA}}(T{)}^{op}$-modules via the derived equivalence $\mathbb{R}Ho{m}_{\text{catA}}(T,-)$.